**Course: **MATH 3110/5110

**Title: **College Geometry

**Section: **100 (Class Numbers 6946/6963

**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

**Academic Year: **2021 - 2022

**Term: **Spring Semester

**Instructor: **Mark Barsamian

**Contact Information: **My contact information is posted on my web page.

**Course Description: **A rigorous course in *axiomatic geometry*. Birkoff's *metric approach* (in which the axioms incorporate the concept of real numbers) is used. Throughout the course, various models will be introduced to illustrate the axioms, definitions and theorems. These models include the familiar *Cartesian Plane* and *Spherical Geometry* models, but also less familiar models such as the *Poincaré Upper Half Plane* and the *Taxicab Plane*. Substantial introduction to the *method of proof* will be provided, including discussion of *conditional statements* and *quantified conditional statements* and their *negations*, and discussion of *proof structure* for *direct proofs*, *proving the contrapositive*, and *proof by contradiction*.

**Prerequisites Shown in Online Course Description: **(MATH 3050 Discrete Math or CS 3000 Introduction to Discrete Structures) and (MATH 3200 Applied Linear Algebra or MATH 3210 Linear Algebra)

**Sufficient Prerequisite: **Concurrent registration in (MATH 3050 Discrete Math or CS 3000 Introduction to Discrete Structures or MATH 3200 Applied Linear Algebra or MATH 3210 Linear Algebra)

**Cross-Listing: **Note that this is a *cross-listed* course: Undergraduate students register for MATH 3110; Graduate students, for MATH 5110.

**Special Needs: **If you have physical, psychiatric, or learning disabilities that require accommodations, please let me know as soon as possible so that your needs may be appropriately met.

**Final Exam Date: **Fri Apr 29, 1:00pm - 3:00pm in Gordy 311

**Syllabus: **This web page replaces the usual paper syllabus. If you need a paper syllabus (now or in the future), unhide the next three portions of hidden content (Textbook, Grading, Course Structure) and then print this web page.

**Textbook Information: **

**Title: ***Geometry: A Metric Approach With Models, 2 ^{nd} Edition*

**Authors: **Millman and Parker

**Publisher: **Springer, 1991

**ISBN Numbers: **

**Softcover ISBN-13:**978-0-387-20139-9**Hardcover ISBN-13:**978-0-387-97412-5

**Supplemental Reading: **Our book relies heavily on logical terminology, terminology that is also widely used in other proof-based math courses. The terminology is sometimes introduced in MATH 3050 and maybe also in CS 3000. You may not have learned that terminology, or you may have learned and forgotten it. The ** Supplemental Reading on Logical Terminology, Notation, and Proof Structure** gives a concise summary of the terminology that is needed for our course.

**Grading: **

During the semester, you will accumulate a * Points Total* of up to

**Attendance:**40 Meetings (Starting Wed Jan 12) @ 1 point each, for 40 points possible**Presentations:**5 Presentations (during Meetings) @ 12 points each, for 60 points possible**Homework:**10 Assignments, 10 points each, for 100 points possible**Best three of the following four items**for a total of 600 points possible:**Quizzes:**Best 8 of 10 quizzes @ 25 points each for a total of 200 points possible**Exam 1:**200 points possible**Exam 2:**200 points possible**Exam 3:**200 points possible

**Final Exam:**200 points possible

At the end of the semester, your * Points Total* will be converted into your

- 900 - 1000 points = 90% - 100% = A-, A = You mastered all concepts, with no significant gaps.
- 800 - 899 points = 80% - 89.9% = B-, B, B+ = You mastered all essential concepts and many advanced concepts, but have some significant gaps.
- 700 - 799 points = 70% - 79.9% = C-, C, C+ = You mastered most essential concepts and some advanced concepts, but have many significant gaps.
- 600 - 699 points = 60% - 69.9% = D-, D, D+ = You mastered some essential concepts.
- 0 - 599 points = 0% - 59.9% = F = You did not master essential concepts..

**There is no curve.**

**Throughout the semester, your current scores and current course grade will be available in an online gradebook in Blackboard.**

**Ziggy: **

**(Optional) Challenge Problems**

We learned about three properties that a **relation on a set** may or may not have: **reflexive, symmetric, transitive**.

Consider strings of three characters, where each character is either *n* or *y*. There are eight possible such strings:

- nnn
- nny
- nyn
- nyy
- ynn
- yny
- yyn
- yyy

For example, the *less than relation*, \(<\), is not reflexive, not symmetric, but it is transitive. This would be denoted by *nny*.

**Challenge Problem #1: **Describe eight relations on the set of *Real Numbers*, \(\mathbb{R}\), that have the above combinations of properties. (Of course, you only need to come up with seven, because I already gave you one.)

On Wed Mar 16, we discussed the *Crossbar Interior of an Angle*. The term is introduced in the book in your **suggested exercise 4.4#25**. You can also read about the term in the meeting topics for Wed Mar 15 (in the **Schedule** farther down this web page). **Logan** had a **Class Presentation** that was relevant to exercise 4.4#25.

**Challenge Problem #2: **Given a **Poincaré angle \(\angle ABC\)**, describe (with a drawing and a mathematical description) the points that are in the *Crossbar Interior of \(\angle ABC\)*.

On Wed Mar 23, in her presentation, **Aoife** drew *Poincaré angle* \(\angle ABC\), for the points \(A=(3,7)\), \(B=(3,4)\), and \(C=(10,3)\). Points \(B,C\) lie on line \(\overleftrightarrow{BC}\) that is described by the symbol \( \ _6L_5\). This line has *center* at \((6,0)\) and missing endpoints at \((1,0)\) and \((11,0)\).

**Mark B** then showed how to find the ** angle bisector** of \(\angle ABC\). The result was that the angle bisector is the

**James** then asked a good question:

The answer to James's question is:

Here is a **general description** of a simpler, **generic problem**:

**The Given Stuff: ** Let \(A=(0,h+1), B=(0,h), C=(2c,h)\), where \(c\) is a *positive real number*. Observe that points \(B\) and \(C\) have the same \(y\) coordinate. This tells us that *Poincaré line* \(\overleftrightarrow{BC}\) will have **center** at the location \((c,0)\).

**Challenge Problem #3: **Prove that the **angle bisector** of *Poincaré angle* \(\angle ABC\) will be a *Poincaré ray* that has endpoint at \(B=(0,h)\) and lies in the *Poincaré line* with **center** at the **right missing endpoint** of \(\overleftrightarrow{BC}\).

**Schedule: **

**Book Sections and Videos:**

- 1.1 Axioms and Models
- 1.2 Sets and Equivalence Relations

**Exercises:**

**Suggested Exercises:**1.2 # 1, 3, 6, 8, 9, 13, 14, 15, 19**Assigned Homework H01**due at the start of the Fri Jan 14 class meeting.**H01 Cover Sheet**

**Mon Jan 10 Meeting Topics**

- Course Organization
- Introduction to Analytic Geometry, Axiomatic Geometry, and Models (Section 1.1 Concepts)
- Introduction to some of the Logical Notation and Terminology that will be used in the course.
- Logical Statements
- Forming New Logical Statements from Old Logical Statements
- The
*Negation* - The
*Logical AND* - The
*Logical OR* - The
*Conditional (If-then) Statement*

- The
- Negating the
*AND,OR*statements:*DeMorgan's Laws* - Negating the
*If-then*statement

**Wed Jan 12 Meeting Topics**

**Topic for today's Meeting Proof: Structure**

There are two common styles of presenting proofs. For students who are learning to write proofs, the style that I do in the videos, with one statement per line and each statement numbered and justified, is a helpful style. These proofs are sometimes referred to as *two-column proofs*. The idea is that the proof could be written into a *two-column table*. Each line of the table would have a *numbered statement* in the *left column*, and a *justification* for that numbered statement in the *right column*.

But in higher level math, it is common to have proofs presented in *paragraph* style, with sentences just strung together into a paragraph. Justifications are also often omitted, especially if the statement is something that a reader in the target audience should be expected to understand without explanation. And certain statements are also often omitted, especially if they are expected to be fairly obvious a reader in the target audience. It is this second style of proof presentation, the *paragraph* style of proof, that our book uses.

The target audience for our book is a reader who has more experience reading and writing proofs than many of the students in MATH 3110/5110. One might think that this would disqualify the book for use in our course. But I feel that the book is an excellent book, one of the best Geometry books that I have seen. And I feel that it is important for students in a 3000-level math course like ours to develop the skill of reading the dense mathematical writing in a book like ours, and to develop the skill of writing proofs. For that reason, I have decided to use our rather advanced book, and to supplement the book with material that will (hopefully) help students in the class develop their reading and proof writing skills.

The Instructional Videos that accompany each section of the textbook are largely about developing the skill of *writing* proofs and, as mentioned above, the proofs created in the videos have the more basic *two-column proof* style. As students work on *writing* proofs in this more basic style, they will certainly become more skilled at *reading* proofs in that basic style. They will also begin to acquire some skill at reading in the more dense *paragraph style* of proof used in the textbook.

But I think it is important to also work directly on the task of improving the students' *reading* skills, so that they can more quickly begin to be able to make sense of the more dense *paragraph style*. For that reason, the *Class Presentations* will also be little drills having to do with making sense of stuff that is in the book.

**Howard Bartels Presentation #1: **(See the **Notes for Video 1.2a**.) In the first problem on **Homework H01**, you are asked to prove the following statement.

For all sets \(A\) and \(B\), if \(A \subset B\) then \(A\cap C \subset B \cap C\)

Show the "*frame*" of the proof. That is, what will need to be the *first* and *last* statements of the proof?

**During the last part of the fri Zuercher Presentation #1: **(See the **Notes for Video 1.2a**.)

Here is a proof that is written in paragraph form, not in numbered statements.

Let \(x\in A \cap \left( B\cup C \right) \). Then \(x\in A\) and \(x\in B \cup C \). Since \(x\in B \cup C \) either \(x\in B\) or \(x \in C\) (or both!). If \(x\in B\) then \(x\in A \cap B\). If \(x\in C\) then \(x\in A \cap C\). Either way, \(x\in \left( A \cap B \right) \cup \left( A \cap C \right) \).

Rewrite this proof as a list of numbered statements, with justifications for each step.

What is the statement that is being proven by the above proof?

**Jingmin Gao Presentation #1: **(See the **Notes for Video 1.2a**.)

Here is a proof that is written in paragraph form, not in numbered statements.

Let \(x\in \left( A \cap B \right) \cup \left( A \cap C \right) \). Then \(x \in \left( A \cap B \right) \) or \(x \in \left( A \cap C \right) \). If \( x\in \left( A \cap B \right) \) then \(x\in A\) and \(x\in B\). Then \(x\in B \cup C\). Therefore, \(x\in A\cap \left(B \cup C\right)\). If \( x\in \left( A \cap C \right) \) then \( x\in A\) and \(x\in C\). Then \(x\in B \cup C\). Therefore, \(x\in A\cap \left(B \cup C\right)\). Either way, \(x\in A\cap \left(B \cup C\right)\).

Rewrite this proof as a list of numbered statements, with justifications for each step.

What is the statement that is being proven by the above proof?

**Jen Shviadok Presentation #1: **(See the **Notes for Video 1.2a**.)

Suppose that a proof has the following structure:

**Proof Part 1:**Prove that \( A \cap \left( B\cup C \right) \subset \left( A \cap B \right) \cup \left( A \cap C \right) \).**Proof Part 2:**Prove that \( \left( A \cap B \right) \cup \left( A \cap C \right) \subset A \cap \left( B\cup C \right) \).

**Fri Jan 14 Meeting Topics**

**James Foley Presentation #1: **(See the **Notes for Video 1.2b**.)

Frick and Frack have been given tasks involving the *relations* presented in **[Examples 5,6]** in the Notes for Video 1.2b

The relation from **[Example 6]** is
$$R = \left\{(x,y) \in \mathbb{R}^2 \middle| x^2=y^2 \right\}$$
and the relation from **[Example 6]** is
$$R = \left\{(x,y) \in \mathbb{R}^2 \middle| x^2+y^2=1 \right\}$$

Frick is asked to prove whether or not the relation in **[Example 5]** is *reflexive*.

Frack is asked to prove whether or not the relation in **[Example 6]** is *reflexive*.

Frick says that the relation in **[Example 5]** is *reflexive* because \(3^2=3^2\), which tells us that \(3\) is related to itself.

Frack says that the relation in **[Example 6]** is *reflexive* because \( \left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2 = 1\), which tells us that \(\frac{1}{\sqrt{2}}\) is related to itself.

Did either give a valid answer? Explain.

**Michael Cooney Presentation #1: **(See the **Notes for Video 1.2b**.)

In your suggested exercise **textbook problem 1.2#9**, you study a relation on the set of all *rectangles*. In that problem, it is discussed that

- The
*height*, \(h\), of a rectangle is defined to be the length of the*longer*of the sides. - The
*width*, \(w\), of a rectangle is defined to be the length of the*shorter*of the sides. - Thus, \(h \ge w > 0\).

The textbook does not introduce a term that would be useful. I'll introduce it, because it is a common term. For a rectangle \(R\) with *height* \(h\) and *width* \(w\), the *aspect ratio* of \(R\) is the ratio
$$\frac{h}{w}$$

A relation is defined on the set of all rectangles as follows.

If rectangle \(R_1\) has height \(h_1\) and width \(w_1\), and rectangle \(R_2\) has height \(h_2\) and width \(w_2\), then we say that the sentence $$R_1 \text{ is related to } R_2$$ which is abbreviated $$R_1 \sim R_2$$ is defined to mean $$\frac{h_1}{w_1}=\frac{h_2}{w_2}$$ This could be spoken as $$\text{the aspect ratio of } R_1 = \text{the aspect ratio of } R_2$$

It turns out that this relation is an *equivalence relation* on the set of all rectangles.

In **textbook problem 1.2#9**, you are asked to *prove* that the relation is an equivalence relation. This would involve three proofs: a proof that the relation is *reflexive*, a proof that the relation is *symmetric*, and a proof that the relation is *transitive*.

Your job in your presentation is *not* to write the three proofs, but rather to just present the "*frame*" of the three proofs. That is, show the *first* and *last* statements of the three proofs. Fill in your frame with one more layer. That is, also show the *second* and *second-to-last* statement of each proof. Leave plenty of empty space for the middle of each of your proofs. Stop there.

**Homework H01** due at the start of the Fri Jan 14 class meeting. **H01 Cover Sheet**

**Quiz Q01** during last part of the Fri Jan 14 class meeting

**Book Sections and Videos:**

- 1.3 Functions
- Section 1.3 of Our book and our course rely heavily on logical terminology. The terminology is also widely used in other proof-based math courses. The terminology is sometimes introduced in MATH 3050 and maybe also in CS 3000. You may not have learned that terminology, or you may have learned and forgotten it. The
gives a concise summary of the terminology that is needed for our course.*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*

**Exercises:**

**Suggested Exercises:**1.3 # 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13**Assigned Homework H02**due at the start of the Fri Jan 21 class meeting.**H02 Cover Sheet**

**Mon Jan 17 is a Holiday: No Class**

**Wed Jan 19 Meeting Topics**

**Nicole Adams Presentation #1: **Your Presentation is a review of some basic logical terminology from your Discrete Math Course, Math 3050. Most MATH 3110 students are very rusty on this stuff, though. If you need to review, here are a couple of possible references:

- See the
**Notes for Video 1.3a** - Look the terminology up in your MATH 3050 book, Discrete Mathematics, by Susannah Epp.
- See Sections I,II,II in the
*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*that I have posted a link to on our course web page. Here is a direct link:.*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*

Let ** Statement S** be the following

- Write the
.*Converse of S* - Write the
.*Contrapositive of S* - Write the
.*Inverse of S* - Write the
.*Negation of S*

We now have five statements:

*Statement S**Converse of S**Contrapositive of S**Inverse of S**Negation of S*

Are any of these statements *logically equivalent*? Explain.

**Mandie Dicicco Presentation #1: **(See the **Notes for Video 1.3a**.) Frack says that to say a function is *one-to-one* means that *for each input, there is exactly one output*. Frack is not correct.

- What
*is*the definition of*one-to-one*? - Frack's answer is a very common incorrect answer. Though it is not the definition of
*one-to-one*, but it is part of the definition of something else. What? (Hint: It is something defined in Section 1.3 and also defined in Video 1.3a.)

**Dani Duey Presentation #1: **(See the **Notes for Video 1.3a**.) Tell the class about the *Natural Exponential* function, \( y=e^{(x)} \).

- Make a large, neat graph of \( y=e^{(x)} \). (You’re welcome to use a computer-generated graph, such as from Desmos.) On your graph, put \( (x,y) \) coordinates on the point with \(x=0 \) and the point with \(x=1 \).
- Use your graph to explain what is the
*domain*of \( y=e^{(x)} \). - Use your graph to explain what is the
*range*of \( y=e^{(x)} \). - Show the
*function diagram*, or*arrow diagram*, for \( y=e^{(x)} \). That is, a diagram of the form $$ e^{( \ \ )}: \_ \_ \_ \_ \rightarrow \_ \_ \_ \_ $$ - Use your graph to explain why \( y=e^{(x)} \) is
*injective*and*surjective*. (This is not a proof. Just explain.) - What is the
*inverse function*for \( y=e^{(x)} \)? What is the*domain*and*range*of the*inverse function*?

**Joe Durk Presentation #1: **(See the **Notes for Video 1.3a** and the **Notes for Video 1.3b** .) In your presentation, you'll be talking about the graph of a *relation* from \( \mathbb{R} \) to \( \mathbb{R} \), and about the graph of its *inverse relation*. (Don't be spooked: there is a lot of writing in the description of your presentation, but you don't actually have that much to do!)

**Part 1: A Relation**

The *equation*
$$y=x^3-6x^2+11x-6$$
defines a *relation* on the set \( \mathbb{R} \). That is, the set of ordered pairs \( (x,y) \) that satisfy the equation is a subset of \( \mathbb{R} \times \mathbb{R} \). That is, a subset of \( \mathbb{R}^2\).

**Task #1: **Make a large, neat graph of the equation \(y=x^3-6x^2+11x-6\). (You’re welcome to use a computer-generated graph, such as from Desmos.) (It might be useful to have a few graphs to work with, because you're going to be drawing on them.)

We could also think of the equation as defining a *relation* from \( \mathbb{R} \) to \( \mathbb{R} \). The *domain* of this relation would be \( \mathbb{R} \), and the *range* of this relation would also be \( \mathbb{R} \). We could call this relation \(f\). The *arrow diagram* for the relation \(f\) would be written
$$ f: \mathbb{R}\rightarrow \mathbb{R}.$$

**Task #2: **Use your graph to explain why the *relation* \(f\) actually qualifies to be called a *function*.

In *function notation* the function would be written
$$f(x)=x^3-6x^2+11x-6.$$
The *arrow diagram*, or *function diagram*, for the *function* \( f \) would be written
$$ f: \mathbb{R}\rightarrow \mathbb{R}$$
which is the same as the arrow diagram for the for the *relation* \( f \).

**Task #3: **Use your graph to explain why the function \(f\) is *surjective*.

**Task #4: **Use your graph to explain why the function \(f\) is *not injective*.

**Part 2: The Inverse Relation**

If we interchange \(x\) and \(y\) in the equation above, we get a new equation.
$$x=y^3-6y^2+11y-6$$
This equation defines a *new relation* on the set \( \mathbb{R} \). That is, the set of ordered pairs \( (x,y) \) that satisfy the equation is a subset of \( \mathbb{R}^2\).

**Task #5: **Make a large, neat graph of the equation \(x=y^3-6y^2+11y-6\). (You’re welcome to use a computer-generated graph, such as from Desmos.)

This new relation is called the *inverse relation* for the relation discussed in **Part 1**. This inverse relation is denoted \(f^{-1}\). The *arrow diagram* for the relation \(f^{-1}\) would be written
$$ f^{-1}: \mathbb{R}\rightarrow \mathbb{R}.$$

**Task #6: **Use your graph to explain why the *inverse relation* \(f^{-1}\) *is not qualified* to be called a *function*.

**Task #7: **Use your graph to find \(f^{-1}(0)\). Explain.

Here's a summary:

- The
*equation*\(y=x^3-6x^2+11x-6\) defines a*relation*from \( \mathbb{R} \) to \( \mathbb{R} \), that can be called \(f\). That is, $$ f: \mathbb{R}\rightarrow \mathbb{R}.$$ The relation \(f\) satisfies the extra requirement to be qualified to be called a*function*from \( \mathbb{R} \) to \( \mathbb{R} \). - The
*equation*\(x=y^3-6y^2+11y-6\) defines a new*relation*from \( \mathbb{R} \) to \( \mathbb{R} \). This new relation is the*inverse relation*for the previous relation. We use the symbol \(f^{-1}\) for the*inverse relation*. But \(f^{-1}\)*does not satisfy*the extra requirement to be qualified to be called a*function*. That is, the relation \(f(x)=x^3-6x^2+11x-6\)*does*have an*inverse relation*, denoted \(f^{-1}\), but it*does not*have an*inverse function*. This is definitely confusing.

**Remark: **Notice that the original equation
$$y=x^3-6x^2+11x-6$$
is solved for \(y\) in terms of \(x\). That is, it defines a *function*
$$f(x)=x^3-6x^2+11x-6.$$
But the equation *cannot* be solved for \(x\) in terms of \(y\). That tells us that there is *no inverse function*.

**Fri Jan 21 Meeting Topics**

**Kelsie Flick Presentation #1: **(See the **Notes for Video 1.3b**.) In your presentation, you'll be talking about two functions:

- Make large, neat graphs of both functions on separate axes. (You’re welcome to use a computer-generated graph, such as from Desmos.) Label all axis intercepts with their \((x,y)\)
*coordinates*and label all asymptotes with their*line equations* - Questions about \(f\)
- What is the
*domain*of function \(f\)? - Using your graph of \(f\), explain what
*range*can be used for function \(f\) so that \(f\) will be bijective. - Summarize your findings by making an
*arrow diagram*for \(f\). That is, a diagram of the form $$f: \_\_\_\_\_ \rightarrow \_\_\_\_\_$$

- What is the
- Questions about \(g\)
- What is the
*domain*of function \(g\)? - Using your graph of \(g\), explain what
*range*can be used for function \(g\) so that \(g\) will be bijective. - Summarize your findings by making an
*arrow diagram*for \(g\).

- What is the
- Questions about
*compositions*. (Some of these questions are*subtle*. Don't be surprised if they are inexplicably hard.)- Find \((g \circ f) (2) \) and \((f \circ g) (2)\). (Show the calculation.)
- Find \((g \circ f) (x) \) and \((f \circ g) (x)\). (Show the calculation.)
- Express \(g \circ f \) and \(f \circ g\) using the notation of the
*identity map*, \(id\). - Based on your result from question (f), what can you say about functions \(f\) and \(g\)? What
*Theorem*allows you to say this? - What is \(id(3)\)? What is \((f \circ g)(3)\)?
- Find \(g(3)\) and then use that result to find \(f(g(3)\). Does your result agree with the result of the previous question? If not, can you explain why?

**Rachel Han Presentation #1: **(See the **Notes for Video 1.3b**.) In your presentation, you'll be talking about properties of functions and properties of compositions of functions.

**(a) **Give an example of functions \(f:S \rightarrow T \) and \(g:T \rightarrow V\) such that \(g \circ f\) is *injective* but \(f\) and \(g\) are *not* both injective.

**(b) **What does your example in (a) tell us about the truth of the following statement? Explain.

In the Notes for Video 1.3b, Barsamian introduces a **Theorem D**

He proves this by proving the *contrapositive* of the statement of Theorem D.

Barsamian also introduces a **Theorem E**

He does not prove this theorem. Your goal is to set up the frame of a proof of Theorem E.

**(c) **Write the *contrapositive* of the statement of Theorem E.

**(d) **To prove Theorem E by proving its *contrapositive*, what would need to be the *frame* of the proof? That is, what would need to be the *first* and *last* statements of the proof? What would need to be the *second* and *second-to-last* statements of the proof? Show the frame of the proof, along with the *second* and *second-to-last* statements of the proof, with a large space in the middle for the gap that needs to be closed. (You don't have to write the full proof: Just stop there.)

**Homework H02** due at the start of the Fri Jan 21 class meeting. **H02 Cover Sheet**

**Quiz Q02** during last part of the Fri Jan 21 class meeting

**Book Sections and Videos:**

- 2.1 Definitions and Models of Incidence Geometry
- 2.2 Metric Geometry
- Our book and our course rely heavily on logical terminology. The terminology is also widely used in other proof-based math courses. The terminology is sometimes introduced in MATH 3050 and maybe also in CS 3000. You may not have learned that terminology, or you may have learned and forgotten it. The
gives a concise summary of the terminology that is needed for our course.*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*

**Exercises:**

**Suggested Exercises For Week 3 and Week 4:**- 2.1 # 1, 3, 5, 6, 8, 10, 11, 12, 13, 16, 18, 19, 24, 25
- 2.2 # 1, 2, 4, 5, 6, 7ii, 9, 10, 11, 12, 17, 18i, 19, 20
- 2.3 # 1, 2, 3, 4, 5, 6

**Assigned Homework H03**due at the start of the Fri Jan 28 class meeting.**H03 Cover Sheet**

**Mon Jan 24 Meeting Topics**

**Kyle Hill (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- Are there any
*incidence geometries*that are not*abstract geometries*? Explain why or why not. - Are there any
*abstract geometries*that are not*incidence geometries*? Explain why or why not. - Is there an
*abstract geometry*in which the set of lines \( \mathscr{L} \) contains just*one*line? Same question for*incidence geometry*. Explain, with justifications or examples.

**Jack Lazenby (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- What is the definition of
*parallel lines*in an*abstract geometry*? - For an abstract geometry \( (\mathscr{P},\mathscr{L}) \), the parallel relation \( || \) can be considered as a relation on the set \( \mathscr{L} \) of lines. Is it an
*equivalence relation*? Explain.

**Taylor Miller (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)
Consider this pair \( (\mathscr{P},\mathscr{L}) \)

- \( \mathscr{P} = \{P,Q,R,S,T,U,V\} \)
- \( \mathscr{L} = \{\{P,Q,R\},\{P,S,U\},\{P,T,V\},\{Q,S,V\},\{Q,T,U\},\{R,S,T\},\{R,U,V\}\} \)

Draw an illustration of \( (\mathscr{P},\mathscr{L}) \). (Hint: Look at book page 25 for some ideas.) Make your drawing large (at least \(4'' \times 4'' \)) and neat, with points labeled with their letters.

- Frick says that \( (\mathscr{P},\mathscr{L}) \) cannot be an
*incidence geometry*because each line contains more than two points. - Frack says that \( (\mathscr{P},\mathscr{L}) \) cannot be an
*incidence geometry*because there are no parallel lines. - Whirlwind says that \( (\mathscr{P},\mathscr{L}) \) is an
*incidence geometry*, but cannot explain why.

Who is correct? Explain.

**Jack Muslovski (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- In an
*abstract geometry*, what does it mean to say that lines \( l_1 \) and \( l_2 \) are*equal*? - In an
*abstract geometry*, what does it mean to say that lines \( l_1 \) and \( l_2 \) are*distinct*? - Why is it that in an
*abstract geometry*, two distinct lines*can*intersect in more than one point, but in an*incidence geometry*two distinct lines*cannot*intersect in more than one point?

**Wed Jan 26 Meeting Topics**

**Logan Prater(Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- Find the
*Cartesian line*through points \(P=(3,4)\) and \(Q=(3,10)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the**Procedure on page 24 of the Notes for Video 2.1a**.) - Graph your
*Cartesian line*. Label your graph well.

(Follow the Presentation Style Guidelines.)

**Sammi Rinicella (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- Find the
*Poincaré line*through points \(P=(3,4)\) and \(Q=(3,10)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the**Procedure on page 26 of the Notes for Video 2.1a**.) - Graph your
*Poincaré line*. Label your graph well.

(Follow the Presentation Style Guidelines.)

**Jake Schneider (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- Find the
*Cartesian line*through points \(P=(3,4)\) and \(Q=(10,3)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the**Procedure on page 24 of the Notes for Video 2.1a**.) - Graph your
*Cartesian line*. Label your graph well.

(Follow the Presentation Style Guidelines.)

**Helen Sitko (Presentation #1): **
(Study Textbook Section 2.1,
Video 2.1a and its Notes,
and
Video 2.1b and its Notes
.)

- Find the
*Poincaré line*through points \(P=(3,4)\) and \(Q=(10,3)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the**Procedure on page 26 of the Notes for Video 2.1a**.) - Graph your
*Poincaré line*. Label your graph well.

(Follow the Presentation Style Guidelines.)

**Fri Jan 28 Meeting Topics**

**Danielle Stevens (Presentation #1): **
(Study Textbook Section 2.2
and Video 2.2a its Notes)

You will be working with points \(P=(3,4)\) and \(R=(10,3)\).

- Find the
*Euclidean distance*, \(D_E(P,Q)\). Show the details of the calculation clearly. - Find the
*Taxicab distance*, \(D_T(P,Q)\). Show the details of the calculation clearly. - Find the
*Max distance*, \(D_S(P,Q)\). Show the details of the calculation clearly. - Illustrate the three distances that you found on a graph showing points \(P\) and \(Q\) and the
*Cartesian line*that passes through them. Label your graph well.

(Follow the Presentation Style Guidelines.)

**Hannah Worthington (Presentation #1): **
(Study Textbook Section 2.2
and Video 2.2a its Notes)

You will be working with points \(P=(3,4)\) and \(Q=(3,10)\) and \(R=(10,3)\).

Following the **procedure on page 26 of the Notes for Video 2.1a**, it can be shown that

- The
*Poincaré line*through points \(P=(3,4)\) and \(Q=(3,10)\) is the*Type I line*\( \ _3L = \left\{(x,y)\in \mathbb{H} \middle| x=3 \right\} \). - The
*Poincaré line*through points \(P=(3,4)\) and \(R=(10,3)\) is the*Type II line*\( \ _6L_5 = \left\{(x,y)\in \mathbb{H} \middle| (x-6)^2+y^2=25 \right\} \).

(You don't have to find those line descriptions: they are given.)

Your job is to do the following three things:

- Find the
*Poincaré distance*, \(D_\mathbb{H}(P,Q)\). Show the details of the calculation clearly. - Find the
*Poincaré distance*, \(D_\mathbb{H}(P,R)\). Show the details of the calculation clearly. - Illustrate the two distances that you found on a graph showing points \(P\), \(Q\), and \(R\) and the two
*Poincaré lines*described above. Label your graph well.

(Follow the Presentation Style Guidelines.)

**Homework H03** due at the start of the Fri Jan 28 class meeting. **H03 Cover Sheet**

**Quiz Q03** during last part of the Fri Jan 28 class meeting.

**Book Sections and Videos:**

- 2.2 Metric Geometry
- 2.3 Special Coordinate Systems (Video 2.3) (Notes for Video 2.3)
- Our book and our course rely heavily on logical terminology. The terminology is also widely used in other proof-based math courses. The terminology is sometimes introduced in MATH 3050 and maybe also in CS 3000. You may not have learned that terminology, or you may have learned and forgotten it. The
gives a concise summary of the terminology that is needed for our course.*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*

**Exercises:**

**Suggested Exercises For Week 3 and Week 4:**- 2.2 # 1, 2, 4, 5, 6, 7ii, 9, 10, 11, 12, 17, 18i, 19, 20
- 2.3 # 1, 2, 3, 4, 5, 6

**Assigned Homework**None

**Mon Jan 31 Meeting Topics**

**Josh Stookey (Presentation #1): **
(Study Textbook Section 2.2,
Video 2.2a its Notes, and
Video 2.2b and its Notes). In particular, study **[Example 1]** on page 19 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(Q=(3,10)\) in the *Poincaré Upper Half Plane*. Last Wednesday, Sammi Rinicella found that the *Poincaré line* containing points \(P\) and \(Q\) is the *type I line* \( \ _3L\). Your job will be to work with the *coordinates* of those points using the *standard ruler* \(f\) for that line in the *Poincaré plane*.

- Find the coordinates of points \(P\) and \(Q\) on the line that passes through them, using the
*standard ruler*\(f\) for the*type I line*\( \ _3L\) in the*Poincaré plane*. Give exact answers in symbols, and decimal approximations. - Compute \( \left|f(P)-f(Q)\right|\). Give an exact answers in symbol, and a decimal approximations.
- Last Friday, Hannah Worthington found that the
*Poincaré distance*between \(P\) and \(Q\) is \(d_\mathbb{H}(P,Q)=\ln\left(\frac{10}{4}\right)\). Compare your result in (b) to Hanna's result. Is the*ruler equation*satisfied? Explain. - Illustrate your results from (a) and (b), along with Hannah's result, \(d_\mathbb{H}(P,Q)=\ln\left(\frac{10}{4}\right)\), using a picture.

(Follow the Presentation Style Guidelines.)

**Nicole Adams (Presentation #2): **
(Study Textbook Section 2.2,
Video 2.2a its Notes, and
Video 2.2b and its Notes). In particular, study **[Example 2]** on page 21 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(R=(10,3)\) in the *Cartesian plane*. Last Wednesday, Jake Schneider found that the *Cartesian line* containing points \(P\) and \(R\) is the *non-vertical line* \(L_{-\frac{1}{7},\frac{31}{7}}\). Your job will be to work with the *coordinates* of those points using the *standard ruler* \(f\) for that line in the *Euclidean plane*.

- Find the coordinates of points \(P\) and \(R\) on the line that passes through them, using the
*standard ruler*\(f\) for the*non-vertical line*\(L_{-\frac{1}{7},\frac{31}{7}}\) in the*Euclidean plane*. Give exact answers in symbols, and decimal approximations. - Compute \( \left|f(P)-f(R)\right|\). Give an exact answers in symbol, and a decimal approximations.
- Last Friday, Danielle Stevens found that the
*Euclidean distance*between \(P\) and \(R\) is \(d_E(P,Q)=\sqrt{50}\). Compare your result in (b) to Danielle's result. Is the*ruler equation*satisfied? Explain. - Illustrate your results from (a) and (b), along with Danielle's result, \(d_E(P,Q)=\sqrt{50}\), using a picture.

(Follow the Presentation Style Guidelines.)

**Michael Cooney (Presentation #2): **
(Study Textbook Section 2.2,
Video 2.2a its Notes, and
Video 2.2b and its Notes). In particular, study **[Example 3]** on page 24 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(R=(10,3)\) in the *Cartesian plane*. Last Wednesday, Jake Schneider found that the *Cartesian line* containing points \(P\) and \(R\) is the *non-vertical line* \(L_{-\frac{1}{7},\frac{31}{7}}\). Your job will be to work with the *coordinates* of those points using the *standard ruler* \(f\) for that line in the *Taxicab plane*.

- Find the coordinates of points \(P\) and \(R\) on the line that passes through them, using the
*standard ruler*\(f\) for the*non-vertical line*\(L_{-\frac{1}{7},\frac{31}{7}}\) in the*Taxicab plane*. Give exact answers in symbols, and decimal approximations. - Compute \( \left|f(P)-f(R)\right|\). Give an exact answers in symbol, and a decimal approximations.
- Last Friday, Danielle Stevens found that the
*Taxicab distance*between \(P\) and \(R\) is \(d_T(P,Q)=8\). Compare your result in (b) to Danielle's result. Is the*ruler equation*satisfied? Explain. - Illustrate your results from (a) and (b), along with Danielle's result, \(d_T(P,Q)=8\), using a picture.

(Follow the Presentation Style Guidelines.)

**Mandie Dicicco (Presentation #2): **
(Study Textbook Section 2.2,
Video 2.2a its Notes, and
Video 2.2b and its Notes). In particular, study **[Example 4]** on page 27 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(R=(10,3)\) in the *Poincaré Upper Half Plane*. Last Wednesday, Helen Sitko found that the *Poincaré line* containing points \(P\) and \(R\) is the *type II line* \( \ _6L_5\). Your job will be to work with the *coordinates* of those points using the *standard ruler* \(f\) for that line in the *Poincaré plane*.

- Find the coordinates of points \(P\) and \(R\) on the line that passes through them, using the
*standard ruler*\(f\) for the*type II line*\( \ _6L_5\) in the*Poincaré plane*. Give exact answers in symbols, and decimal approximations. - Compute \( \left|f(P)-f(R)\right|\). Give an exact answers in symbol, and a decimal approximations.
- Last Friday, Hannah Worthington found that the
*Poincaré distance*between \(P\) and \(R\) is \(d_\mathbb{H}(P,R)=\ln(6)\). Compare your result in (b) to Hanna's result. Is the*ruler equation*satisfied? Explain. - Illustrate your results from (a) and (b), along with Hannah's result, \(d_\mathbb{H}(P,R)=\ln(6)\), using a picture.

(Follow the Presentation Style Guidelines.)

**Wed Feb 2 Meeting Topics**

**Dani Duey (Presentation #2): **
(Study Textbook Section 2.2 and Video 2.2b and its Notes. In particular, study **[Example 5]** on page 33 of the notes for Video 2.2b.)

Find the point \(P\) on the *non-vertical line* \(L_{-\frac{1}{2},5}\) that has coordinate \( \lambda = 2 \) in the *standard ruler*
for that line in the *Euclidean plane*. Illustrate your result.
(Study **[Example 2]** on page 21 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines.)

**James Foley (Presentation #2): **
(Study Textbook Section 2.2 and Video 2.2b and its Notes. In particular, study **[Example 7]** on page 35 of the notes for Video 2.2b.)

Find the point \(P\) on the *type II line* \( \ _0L_5 \) that has coordinate \( \lambda = \ln{(2)} \) in the *standard ruler*
for that line in the *Poincaré plane*. Illustrate your result.
(Study **[Example 4]** on page 27 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines.)

**Joe Durk (Presentation #2): **
(Study Textbook Section 2.3 and Video 2.3 and its Notes. In particular, study **[Example 1]** on page 24 of the notes for Video 2.3.)

In the *Euclidean plane*, let \(A=(4,3)\) and \(B=(0,5)\). Find a *ruler with \(A\) as origin and \(B\) positive.*. Illustrate your result.
(Study **[Example 2]** on page 21 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines.)

**Kelsie Flick (Presentation #2): **
(Study Textbook Section 2.3 and Video 2.3 and its Notes. In particular, study **[Example 3]** on page 29 of the notes for Video 2.3.)

In the *Poincaré plane*, let \(A=(4,3)\) and \(B=(0,5)\). Find a *ruler with \(A\) as origin and \(B\) positive.*. Illustrate your result.
(Study **[Example 4]** on page 27 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines.)

**Fri Feb 4 Classes Cancelled Because of Weather**

**Book Sections and Videos:**

- 3.1 An Alternative Description of the Cartesian Plane
- 3.2 Betweenness
- Our book and our course rely heavily on logical terminology. The terminology is also widely used in other proof-based math courses. The terminology is sometimes introduced in MATH 3050 and maybe also in CS 3000. You may not have learned that terminology, or you may have learned and forgotten it. The
gives a concise summary of the terminology that is needed for our course.*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*

**Exercises:**

**Suggested Exercises For Week 5:**- 3.1#1, 5, 7, 8, 9b
- 3.2 # 1, 2, 3, 5, 7, 8, 9, 10, 13

**Assigned Homework Due During Week 5:**None

**Exam X1 (Covering Chapters 1,2) for the full duration of the Mon Feb 7 class meeting**

**No Calculators (They are not needed, anyway.)**

**No Phones**

**No Books or Notes**

The Exam is 6 problems, 6 pages, printed on front & back of three sheets of paper.

(A fourth sheet of paper has definitions and formulas on both sides.)

- (30 points) Question about
*Relations*(concepts from Section 1.2) - (40 points) Question about
*Abstract*and*Incidence Geometry*and*Parallel Lines*(concepts from Section 2.1) - (30 points) Prove or disprove a fact about
*Properties of Functions*(concepts from Section 1.3) - (40 points) Question about
*Rulers*in the*Euclidean Plane*(concepts from Section 2.2, 2.3) - (30 points) Prove a fact about
*Abstract Geometry*(concepts from Section 2.1) - (30 points) Prove a fact about
*Metric Geometry*(concepts from Section 2.2)

**Observe that Exam X1 will not have any problems involving computing line equations or finding rulers and coordinates for Poincaré lines.** You had a good dose of that on Homework H03 and Quiz Q03, and you'll get plenty more of it in coming weeks. (Bwahahaha!) So save time and don't bother reviewing

**Wed Feb 9 Meeting Topics**

**Topic for Today:
An Alternative Description of the Cartesian Plane (using Vectors)
(concepts from Section 3.1)**

**Mark B: **Review the idea of *Vector Space over a Field*, and introduce the vector space \((\mathbb{R}^2,+,scalar mult)\) over the field \(\mathbb{R}\).

The basic vector space operations of *vector addition* and *scalar multiplication* are defined in the book on page 42, in **Definition** parts (i) and (iii).

For vectors \( A=(x_A,y_A ) \) and \( B=(x_B,y_B ) \),

- Vector Addition: \( A+B=(x_A+x_B,y_A+y_B ) \)
- Scalar Multiplication: \( r=(rx_A,ry_A) \)

The definition there also introduces three more operations involving vectors:

- Vector Subtraction: \( A+B=A+(-1)B=(x_A-x_B,y_A-y_B ) \)
- The Inner Product of Two Vectors: \( \left< A,B \right> = x_Ax_B+y_A y_B \)
- The Norm of a Vector: \( \left \| A \right \| =\sqrt{\left< A,A \right> }=\sqrt{x_A x_A+y_A y_A }=\sqrt{x_A^2+y_A^2 } \)

In **Proposition 3.1.1** on page 43 of the textbook, a bunch of facts about the above *Vector Space Operations* are presented. In some sense, these facts look familiar: they resemble some properties of *arithmetic* such as the *commutative*, *associative*, or *distributive* properties. But these facts are *not* about ordinary *arithmetic*; they are about *Vector operations*. For the reader, it is worthwhile to take the time to *prove* some of these facts, so that the reader will better understand what they are looking at.

In your **Homework H04**, due this coming Monday, you will be asked to do that. Today, James will prove one of the facts as his *presentation*.

**James Foley, Rachel Han, and Kyle Hill:** Read textbook Section 3.1. In particular, read the definition of the vector space \( \mathbb{R}^2 \) on page 42. You will need to understand the meaning of the symbols for *vector addition*, *scalar multiplication*, *inner product*, and *norm*, introduced in the book and reviewed above.

**James Foley (Presentation #2): **(Preparation: Read the first two pages of textbook Section 3.1. In particular, read the definition of the vector space \( \mathbb{R}^2 \) on page 42. You will need to understand the meaning of the symbols for *vector addition* and *inner product*, introduced in the book and reviewed above.)

Present a proof of Proposition 3.1.1(vii): $$ \langle A+B,C \rangle = \langle A,C \rangle + \langle B,C \rangle $$ Set up your proof in the following format:

- Show the calculation of the
*left side*. - Show the calculation of the
*right side*. - Confirm that the
*left side*equals the*right side*.

(Follow the Presentation Style Guidelines.)

**Mark B: **A couple pages later, on pages 45 and 46 of the textbook, two *inequalities* are presented that are satisfied by vectors in the vector space \(\mathbb{R}^2\). The **Cauchy-Schwarz Inequality** is presented in **Proposition 3.1.5**, and the **Triangle Inequality for the Norm** is presented at the top of page 46, during the proof of **Proposition 3.1.6**.

- The
*Cauchy-Schwarz Inequality*: \( \left\vert \left< A,B \right> \right \vert \leq \left \| A \right \| \cdot \left \| B \right \| \) - The
*Triangle Inequality for the Norm*: \( \left \| A+B \right \| \leq \left \| A \right \| + \left \| B \right \| \)

It is useful to compute some *examples involving actual vectors*, to see how these inequalities are actually satisified in those examples. In your **Homework H04**, due this coming Monday, you will be asked to do that. Today, Rachel and Kyle will do two such examples in their *presentations*.

**Rachel Han and Kyle Hill: ** (Preparation: Read the first two pages of textbook Section 3.1. In particular, read the definition of the vector space \( \mathbb{R}^2 \) on page 42. You will need to understand the meaning of the symbols for *vector addition*,*scalar multiplication*, the *inner product*, and the *norm*, introduced in the book and reviewed above.) Your job is to present some examples showing how all vectors satisfy the *Cauchy-Schwarz Inequality* and the *Triangle Inequality for the Norm*.

- The
*Cauchy-Schwarz Inequality*: \( \left\vert \left< A,B \right> \right \vert \leq \left \| A \right \| \cdot \left \| B \right \| \) - The
*Triangle Inequality for the Norm*: \( \left \| A+B \right \| \leq \left \| A \right \| + \left \| B \right \| \)

**Rachel Han (Presentation #2):**Let \(A=(2,1)\) and \(B=(4,5)\)**Kyle Hill (Presentation #2):**Let \(A=(2,1)\) and \(B=(-1,2)\)

- Draw your vectors in the \(x,y\) plane. Make your drawing big and neat.
- Compute the following:

\( \left< A,B \right> \), \( \left\vert \left< A,B \right> \right \vert \), \( \left \| A \right \| \), \( \left \| B \right \| \), \( \left \| A \right \| \cdot \left \| B \right \| \)

Is the*Cauchy-Schwarz Inequalilty*satisfied? Explain, using the following format- Show the calculation of the
*left side*. - Show the calculation of the
*right side*. - Confirm that
*left side*\( \leq \)*right side*.

- Show the calculation of the
- Compute the following:
\(A+B\),
\( \left \| A+B \right \| \),
\( \left \| A \right \| + \left \| B \right \| \)

Is the*Triangle Inequality for the Norm*satisfied? Explain, using the same format that you used to verify the previous inequality.

(Follow the Presentation Style Guidelines.)

**Mark B: **The reason that the authors introduce vectors and vector operations in Section 3.1 is that some calculations and proofs involving the *Cartesian Plane* can be simplified if vector operations are used. The authors start by using vector calculations to describe the *line* through distinct points \(P\) and \(Q\). They denote the line by \(L_{AB}\).

**Jack Lazenby (Presentation #2): ** (See Section 3.1 and Video 3.1) Let \(P=(4,2) \)and \(Q=(6,7) \).

- Use vector notation to build a description of line \( \overleftrightarrow{PQ} \).
- Illustrate your line \( \overleftrightarrow{PQ} \) along with vectors \( P \) and \( Q-P \).
- Show how the initial presentation of \( \overleftrightarrow{PQ} \) involving
*vector calculations*can be simplified to a final presentation of \( \overleftrightarrow{PQ} \) that is a*parametric form*of the description of the line.

(Follow the Presentation Style Guidelines.)

**Fri Feb 11 Meeting Topics**

**Topic for Today: Betweenness
(concepts from Section 3.2)**

**Mark B: **In Section 3.2, the authors introduce the concept of **Betweenness** for points in a *metric geometry*. The concept is be defined in terms of *collinearity of points* and the *distance between points*. *After* the authors define *betweenness for points*, they then proceed to define

In my Video 3.2a, I prefer to *first* introduce *betweenness for real numbers*, and then introduce

(Mark B review Video 3.2a pages 2 - 11.)

Having discussed the concept of *betweenness for real numbers*, my Video 3.2a then goes on to introduce

(Mark B review Video 3.2a pages 12 - 13.)

In the textbook and in my Video 3.2a, immediately after *betweenness for points* is defined, there are basic examples involving the definition. In your Homework H04 problems [4],[5], you will be asked to do some basic problems involving the definition. Taylor, Jack, Logan, and Sammi will present examples involving the definition.

**Taylor Miller (Presentation #2): **Consider the points \( A=(2,3) \), \( B=(6,5) \), and \( C=(8,6) \) in the *Euclidean plane*.

- Draw the points \( A,B,C \).
- Are the points \( A,B,C \)
*collinear*? Explain. (If you say the points are collinear, add the line to your drawing and label the line with its equation.) - Compute the quantities \( d_E(A,B) \), \( d_E(B,C) \), and \( d_E(A,C) \).
- Is the equation \( d_E(A,B) + d_E(B,C) = d_E(A,C) \) true?
- Is \( B \)
*between*\(A\) and \( C \) in the*Euclidean plane?*Explain.

(Follow the Presentation Style Guidelines.)

**Jack Muslovski (Presentation #2): **Consider the points \( A=(2,3) \), \( B=(6,5) \), and \( C=(8,6) \) in the *Taxicab plane*.

- Draw the points \( A,B,C \).
- Are the points \( A,B,C \)
*collinear*? Explain. (If you say the points are collinear, add the line to your drawing and label the line with its equation.) - Compute the quantities \( d_T(A,B) \), \( d_T(B,C) \), and \( d_T(A,C) \).
- Is the equation \( d_T(A,B) + d_T(B,C) = d_T(A,C) \) true?
- Is \( B \)
*between*\(A\) and \( C \) in the*Taxicab plane?*Explain.

(Follow the Presentation Style Guidelines.)

**Logan Prater (Presentation #2): **Consider the points \( A=(2,3) \), \( B=(7,4) \), and \( C=(8,6) \) in the *Taxicab plane*.

- Draw the points \( A,B,C \).
- Are the points \( A,B,C \)
*collinear*? Explain. (If you say the points are collinear, add the line to your drawing and label the line with its equation.) - Compute the quantities \( d_T(A,B) \), \( d_T(B,C) \), and \( d_T(A,C) \).
- Is the equation \( d_T(A,B) + d_T(B,C) = d_T(A,C) \) true?
- Is \( B \)
*between*\(A\) and \( C \) in the*Taxicab plane*? Explain.

(Follow the Presentation Style Guidelines.)

**Sammi Rinicella (Presentation #2): **Consider the points \( A=(0,5) \), \( B=(33,32) \), and \( C=(70,73) \) in the *Euclidean plane*.

Frick, Frack, and Moonbeam have been asked to determine whether \(B\) is *between* \(A\) and \(C\).

Frick decides to start by making a graph.

**Sammi: **Present a graph of line \( \overleftrightarrow{AC} \) along with points \( A,B,C \).

Frick observes that it is pretty obvious from the graph that \(B\) is *between* \(A\) and \(C\).

Frack decides to check distances. He says that

- \( d_E(A,B) = 45.967 \)
- \( d_E(B,C) = 51.624 \)
- \( d_E(A,C) = 97.591\)
- The sum \(45.967+51.624=97.591\). That is, \( d_E(A,B) + d_E(B,C) = d_E(A,C) \)
- Therefore \( B \) is
*between*\(A\) and \( C \) in the*Euclidean plane*.

Moonbeam is suspicious of Frick's graph, because it is about \(2" \times 2"\). So she decides to check *slopes* to verify collinearity of the points. She says that

- The slope of line \( \overleftrightarrow{AB} \) is \(m=0.97\).
- The slope of line \( \overleftrightarrow{BC} \) is \(m=0.97\).

It turns out that all three are **wrong**! Point \( B \) is *not between* \(A\) and \( C \) in the *Euclidean plane*.

How can they be wrong? Explain.

(Follow the Presentation Style Guidelines.)

**Mark B: **Having defined *betweenness for real numbers* and

(Mark B review pages 5 - 8 in Video 3.2b.)

The most important property of *betweenness for points* on a line \(l\) is the fact that it is related to the

**Equivalence Theorems**

It is very important to understand the style of presentation that I use for Theorem 3.2.3, and the style that the *textbook* uses for the same theorem.

(Mark B compare Theorem 3.2.3 on page 6 of Video 3.2b with Theorem 3.2.3 on page 49 of the book.)

In general, a common style of proof statement is the ** if and only if** statement (the style used by the textbook).

*Statement A if and only if Statement B*

Understand that what this really means is the following:

*Statement A is true if and only if Statement B is true*

In other words, Statements A and B are either both true, or both false.

I prefer to present such theorems in a different style, a style that I call ** Equivalence Theorems**.

*The Following Are Equivalent (TFAE):
*

- Statement A
- Statement B

Understand that what this really means is the following:

*The Following Are Equivalent (TFAE):
*

- Statement A is true.
- Statement B is true.

Again, this simply means that Statements A and B are either both true, or both false.

It is also crucial that you undertand that an *if and only if* theorem, or an *Equivalence* theorem, *does not* tell you that either of the statements is true! It only tells you that either both statements are true, or they are both false. It is common for students to misuse *if and only if* theorems, or *Equivalence* theorems, using them to claim, out of the blue, that some statement is true. Maybe I'll illustrate this common mistake with an example of an *Equivalence* theorem and its misuse.

**The Joe Burrow Theorem**
*The Following Are Equivalent (TFAE):
*

- A person's teammate is Joe Burrow.
- A person plays for the Cincinnati Bengals.

Consider the following:

*Mark Barsamian plays for the Cincinnati Bengals (by the Joe Burrow Theorem).*

That's nonsense. I have a hard time bending over to tie my shoe. The Joe Burrow Theorem cannot be used to say, out of the blue, that I play for the Cincinnati Bengals.

But consider the following:

*Trayveon Williams's teammate is Joe Burrow (by the Joe Burrow Theorem (ii) \(\rightarrow\) (i)).*

That statement is true, and it has a valid justification.

Observe that to *use* an *if and only if* theorem, or an *Equivalence* theorem, one must *already know* that one of the statements mentioned in the theorem is *true*. And when one cites the theorem as a justification for a line in a proof, one should specify which statement is *already known to be true* and which statement *is being declared true by using the theorem*. (See the justification for the statement about Trayveon Williams.)

(Of course, it could be that you know that one of the statements mentioned in an *Equivalence Theorem* is *false*. Then the *Equivalence Theorem* would be used to prove that the *other* statement mentioned in the theorem is *also false*.)

**Example about Betweenness for Points on a Line: ** Suppose that points \(A,B,C\) are collinear on some line \(l\) in a metric geometry, and suppose that using a ruler \(f\) for line \(l\), the coordinates of the three points are \(f(A)=17\) and \(f(B)=12\) and \(f(C)=9\).

- Prove that \(B\)
*is between*\(A\) and \(C\). Do two different proofs:**Proof #1:**Do the proof just using the*Definition of Betweenness for Points*.**Proof #2:**Do the proof again, this time using the*Definition of Betweenness for Real Numbers*and**Theorem 3.2.3**.

- Draw a diagram to illustrate the configuration of points and their coordinates.

**Two More Theorems about Betweenness of Points**

**Proposition 3.2.5**, presented on page 50 of the textbook and on page 9 of the Notes for Video 3.2b, is about ** Betweenness of Points Expressed Using the Vector Description of a Line**. I do a partial proof of the proposition in the video. Here, I will just illustrate why the theorem makes sense. Consider points \(B\) of the form
$$B=A+t(C-A)$$

- Consider where point \(B\) will be when the parameter \(t\) has the value \(t=0\).
- Consider where point \(B\) will be when the parameter \(t\) has the value \(t=1\).
- Consider where point \(B\) will be when the parameter \(t\) has a value \(0 < t < 1\).

**Proposition 3.2.6**, presented on page 50 of the textbook and on page 12 of the Notes for Video 3.2b, is about ** Existence of Points with Certain Betweenness Relationships**. I do a proof of the theorem in the video. In your Homework H04, you will be asked to

- Illustrate Claim (i): Given distinct points \(A,B\) in a metric geometry, there exists a point \(C\) with \(A-B-C\).
- Illustrate Claim (ii): Given distinct points \(A,B\) in a metric geometry, there exists a point \(D\) with \(A-D-B\).

**Book Sections and Videos:**

- 3.3 Line Segments and Rays
- 3.4 Angles and Triangles
- Our book and our course rely heavily on logical terminology. The terminology is also widely used in other proof-based math courses. The terminology is sometimes introduced in MATH 3050 and maybe also in CS 3000. You may not have learned that terminology, or you may have learned and forgotten it. The
gives a concise summary of the terminology that is needed for our course.*Supplemental Reading on Logical Terminology, Notation, and Proof Structure*

**Exercises:**

**Suggested Exercises For Week 6:**- 3.3 # 2, 3, 4, 9, 11, 12, 13, 14, 15
- 3.4 # 1, 2, 3, 4

**Assigned Homework Due During Week 6: H04**due at the start of the Mon Feb 14 class meeting.**H04 Cover Sheet**

**Mon Feb 14 Meeting Topics**

**Homework H04** due at the start of the Mon Feb 14 class meeting. **H04 Cover Sheet**

**Quiz Q04** during last part of the Mon Feb 14 class meeting.

**Wed Feb 16 Meeting Topics**

**Fri Feb 18 Meeting Topics**

**One Leftover Topic from Section 3.2**

**Mark B: **We discussed **Section 3.2 Betweenness** last week. The concept comes up in one of your suggested exercises for this week, so it is worth revisiting the concept. **Jake** will do a presentation involving an exercise of the following type:

**Jake Schneider (Presentation #2): **
(Review the definition of ** Betweenness for Points**, found on page 47 of the book and on page 13 of the Notes for Video 3.2a.)
Suppose that \(A,P,X\) are

- \(AP=5\)
- \(PX=8\)
- \(AX=3\)

**One More Topic From Section 3.3: Midpoint of a Line Segment.**

**Mark B: **The textbook puts the definition of a ** midpoint** of a line segment in Exerise 3.3#11 on page 58. It can also be found on page 19 of the Notes for Video 3.3b.

Observe that I have drawn the midpoint \(M\) so that \(A-M-B\), but nothing in the definition of *midpoint* specifies that the points \(A,M,B\) have that betweenness relationship. **Helen** will show in her presentation that the betweenness relationship is true.

**Helen Sitko (Presentation #2): **(Study pages 19-20 of the Notes for Video 3.3b.) (Suggested Exercise 3.3#11a) Prove the following:

If \(M\) is a midpoint of \(\overline{AB}\) then \(A-M-B\).

**Section 3.4: Angles and Triangles**

**Mark B: **The textbook definitions of ** angle** and

**Question for the Class: **Frick and Frack are co-teaching a high school geometry class. Today, they are discussing *angles*.

- Frick says that \( \angle ABC = \angle BCA \) because as long as you keep the three points in the same order, the angle doesn't change.
- Frack says that \( \angle ABC = -\angle CBA \) because one is clockwise and the other is counter-clockwise.

**Danielle Stevens (Presentation #2): **
(Study Textbook Section 3.4 and Video 3.4 and the Notes for Video 3.4.)
Let \(A=(2,10)\) and \(B=(2,6)\) and \(C=(10,6)\).

- Draw
*Euclidean*\( \angle ABC \). - Draw
*Poincaré*\( \angle ABC \).

(Follow the Presentation Style Guidelines.)

**Hannah Worthington (Presentation #2): **
(Study Textbook Section 3.4 and Video 3.4 and the Notes for Video 3.4.)
Let \(A=(-12,5)\) and \(B=(0,13)\) and \(C=(12,5)\).

- Does
*Euclidean*\( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist - Does
*Euclidean*\( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist - Does
*Poincaré*\( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist - Does
*Poincaré*\( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist

(Follow the Presentation Style Guidelines.)

**During the last part of the fri Zuercher (Presentation #2): **
(Study Textbook Section 3.4 and Video 3.4 and the Notes for Video 3.4.)
Let \(A=(-6,4)\) and \(B=(0,4)\) and \(C=(6,4)\).

- Does
*Euclidean*\( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist - Does
*Euclidean*\( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist - Does
*Poincaré*\( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist - Does
*Poincaré*\( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist

(Follow the Presentation Style Guidelines.)

**Question for the Class: **Frick and Frack are discussing *triangles* with their high school geometry class.

- Frick says that \( \Delta ABC = \Delta BCA \) because as long as you keep the three vertices in the same order, the angle doesn't change.
- Frack says that \( \Delta ABC = -\Delta ACB \) because the vertices changed order.

**Book Sections and Videos:**

- 4.1 The Plane Separation Axiom (Video 4.1) (Notes for Video 4.1)
- 4.2 PSA for the Euclidean and Poincaré Planes (Video 4.2) (Notes for Video 4.2)
- 4.3 Pasch Geometries (Video 4.3) (Notes for Video 4.3)

**Exercises:**

**Suggested Exercises For Week 7:**- 4.1 # 1, 2, 4, 5, 6, 8, 9, 10, 11, 13
- 4.2 # 1, 4
- 4.3 # 1, 2, 3, 4, 7

**Assigned Work Due During Week 7:****Homework H05**due at the start of the Mon Feb 21 class meeting.**H05 Cover Sheet****Take-Home Quiz Q05**due at the start of the Mon Feb 21 class meeting.

**Assigned Homework Due During Week 8: H06**due at the start of the Mon Feb 28 class meeting.**H06 Cover Sheet**

**Mon Feb 21 Meeting Topics**

**Mark B: **We could consider one goal of our course in *Axiomatic Geometry* to be to come up with an *axiom system* that precisely describes, in words, the kind of behavior that we are used to seeing in the drawings that we have made all our lives. That is, we want to come up with an *axiom system* that describes all of the behavior of the *Euclidean Plane*

It is worthwhile to review the three axiom systems for geometry that we have encountered so far, and consider how they have done at fulfilling that goal.

- Abstract Geometry (introduced on page 17 of the book and on page 3 of the Notes for Video 2.1a)
- Incidence Geometry (introduced on page 22 of the book and on page 3 of the Notes for Video 2.1b)
- Metric Geometry (introduced on page 30 of the book and on page 15 of the Notes for Video 2.2b)

Notice that after starting with *Abstract Geometry*, we could consider each successive geometry as an *improvement* in the sense that it specified (in additional axioms and definitions) something that had not been specified in the previous geometry. In this way, each successive geometry describes more precisely how the geometry has to behave.

A natural question, is:

That is, does the axiom system for *Metric Geometry* fully specify the behavior of the straight line drawings that we are used to drawing? After all, we have seen that *The Euclidean Plane* is a *model* of *metric geometry*.

If you consider that we have seen three *other* models of *metric geometry* (the *Taxicab plane*, the *Max plane*, the *Poincaré Upper Half Plane*), then you will realize that *we are not done*. That is, the axiom system for *metric geometry* allows for some pretty weird behavior.

But what additional axioms will we need? Obvious *weird* behavior that we have seen in *metric* geometries includes

*Circles*in the*Taxicab plane*and*Max plane*that look like what we think of as*squares*.*Lines*in the*Poincaré Upper Half Plane*that look like what we think of as*semicircles*.

Well, that's exactly what we will be doing in the next couple of months. What's interesting, though, is that we will start by adding an axiom that *does not* rule out those three weird models. In fact at first, it might not be clear that we even *need* this new axiom. The new axiom has to do with what is called *Plane Separation*, which is the subject of Chapter 4.

**Chapter 4 Plane Separation**

**Section 4.1 The Plane Separation Axiom**

Consider the four familiar behaviors of drawings that are presented on page 2 of the Notes for Video 4.1. We could think of those four behaviors as *desirable plane separation behavior*. We certainly want our axiom system for axiomatic geometry to be specific enough, complete enough, that it guarantees that this sort of desirable behavior will happen in our axiomatic geometry.

A natural question is:

Interestingly (this is proven later in Chapter 4), all four of our familiar models of *metric geometry* (Euclidean, Taxicab, Max, Poincaré) do exhibit this behavior. Those facts can be proven in *calculations* involving those four models.

But if you try to prove theorems saying that all *metric geometries* have that desirable plane separation behavior, you will be frustrated. It turns out that it is *impossible* to prove that all *metric geometries* have that desirable plane separation behavior, because there are examples of *metric* geometries that *do not* have that behavior. (We will study one, called the *Missing Strip Plane*, this week.)

So if we want to insure that our axiomatic geometry *does* have that desirable plane separation behavior, we will have to add an *axiom* that articulates that requirement. That axiom is called the *Plane Separation Axiom*.

Before we can understand the wording of that axiom, though, we will need to understand some new terminology: *Partition of a Set*, and *Convex Sets*.

**Partition of a Set**

The Definition of a * Partition of a Set* is presented on page 4 of the notes for Video 4.1. The terminology is not used in the book, but it should be, because the underlying concepts are used in the

**Convexivity**

The Definition of a * Convexivity* is presented

**Howard Bartels Presentation #2: **(Study Book Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 pages 1 - 14)

The goal is to prove this theorem:

- Make a drawing that illustrates the statement of the theorem. (See page 6 of the video notes for an example of how such a drawing can look.)
- Prove the theorem.

The theorem that Howard proved involved a specific, known set (a *line*) and the proof was fairly simple.

On page 10 of the notes for Video 4.1, a more abstract theorem is presented and proved:

You can see that the proof is very straightforward: all of the steps really come from considering *proof structure*.

But some simple sounding statements about convexivity can be a nuisance to prove. Jingmin will consider such a statement, and will make drawings to illustrate the statement.

**Jingmin Gao Presentation #2: **(Study Book Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 pages 1 - 14)

Our goal is to prove this theorem:

You don't have to prove the theorem--we'll discuss it as a class. Your job is just to make some drawings that illustrate the statement of the theorem. (See page 6 of the video notes for an example of how such a drawing can look.) Your drawing task is made a little harder by the fact that there are different configurations that the sample points \(A,B\) can have on ray \( \overrightarrow{PQ} \). Make drawings that illustrate the statement of the theorem for the different possible configurations of points \(A,B\) can have on ray \( \overrightarrow{PQ} \).

**Mark B: **Jingmin's illustrations show that a proof of the statement that every *ray* is a convex set will have to involve a bunch of different cases. I will prove just *one* of the cases here, to give you an idea of what a nuisance such a proof would be.

But once we prove (or have a sense of how to prove) certain statements that are hard to prove, we can use those proven statements to prove other stuff. For example, consider this theorem:

One might think that a proof of this statement would be a nuisance, involving a few different cases, as happened in the proof of Jingmin's theorem about *rays* being convex. But it turns out that there is a very simple proof involving previously proved statements:

**First Proof of Mark's Theorem about Segments:**

- Suppose that a segment \(\overline{AB}\) is given.
- We know that \(\overline{AB} = \overrightarrow{AB}\cap\overrightarrow{BA} \) (by the result of book exercise 3.3#15).
- The rays \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) are
*convex*(by**Jingmin's Theorem**). - Therefore, the intersection \(\overline{AB} = \overrightarrow{AB}\cap\overrightarrow{BA} \) will be a
*convex set*(by the**Theorem from Exercise 4.1#1**, discussed earlier).

**Vacuously True Statements**

Consider this theorem.

It is possible to prove this theorem with a proof similar to the proof that we just saw for segments:

**Proof of Mark's Theorem about Points:**

- Suppose that a point \(P\) is given.
- We know that there exist at least two distinct lines \(l\) and \(m\) that contain \(P\) (by the
**Two-Line Theorem of Incidence Geometry**, proven on Homework H04). - The lines \(l\) and \(m\) are
*convex*(by**Howard's Theorem**(from Exercise 4.1#2)). - Therefore, the intersection \(P = l \cap m \) will be a
*convex set*(by the**Theorem from Exercise 4.1#1**, discussed earlier).

This proof certainly works, but notice that it relies on three previously-proven theorems, theorems whose proofs you might have forgotten:

- The
**Two-Line Theorem of Incidence Geometry**was assigned to be proven on Homework H04, but most of you had trouble with that proof. **Howard's Theorem**required a proof that Howard supplied.- The
**Theorem from Exercise 4.1#1**had to be proven. (I proved it in Video 4.1.)

It is possible (and much better) to give a *simpler proof* of **Mark's Theorem about Points**, a proof that does not require that you remember so many previous theorems. To discover the simpler proof, it is helpful to remind ourselves of what it means to say that *a point is a convex set*, and to think about what would have to happen for that statement to be *false*.

To say that

means the following *universally-quantified* statement, which we can call *Statement S*

The negation of this statement is the following *existentially-quantified* statement:

Think about it: There is no pair of points \(A,B\) with \(A\neq B\) such that \(A,B \in P\). So it is impossible for * the Negation of Statement S* to be true.

Since the * Negation of Statement S* cannot possibly be true, we realize that

Observe that ** Statement S** is true because

I can say that the statement is true, because the only way for the statement to be false would be for there to be an elephant in the room that is *not* purple. Since there are no objects that exist that could cause the statement to be false, we say that the statement is * vacuously true*.

**Homework H05** due at the start of the Mon Feb 21 class meeting. **H05 Cover Sheet**

**Take-Home Quiz Q05** due at the start of the Mon Feb 21 class meeting.

**Wed Feb 23 Meeting Topics**

**Section 4.1 The Plane Separation Axiom, continued**

On Monday, we discussed four familiar behaviors of drawings that are presented on page 2 of the Notes for Video 4.1. We could think of those four behaviors as *desirable plane separation behavior*. We certainly want our axiom system for axiomatic geometry to be specific enough, complete enough, that it guarantees that this sort of *desirable plane separation behavior* in our axiomatic geometry.

I discussed the fact that although all four of our familiar models of *metric geometry* (Euclidean, Taxicab, Max, Poincaré) do exhibit this behavior, there are examples of *metric* geometries that *do not* have that desirable behavior. (We will study one, called the *Missing Strip Plane*, this week.) So, if we want to insure that our axiomatic geometry *does* have that *desirable plane separation behavior*, we will have to add an *axiom* that articulates that requirement. The axiom that will be added is called the *Plane Separation Axiom*.

On Monday, I mentioned that before we can understand the wording of that axiom, though, we will need to understand some new terminology: *Partition of a Set*, and *Convex Sets*. So on that day,

- we discussed the definition of
**Partition of a Set**, which is presented on page 4 of the notes for Video 4.1. - and we discussed the definition of
**Convex Set**, which is presented - and we discussed a few theorems about Convex Sets.
**Howard's Theorem (from Exercise 4.1#2):***In any metric geometry, every**line*is a*convex set*.**Jingmin's Theorem:***In any metric geometry, every**ray*is a*convex set*.**Mark's Theorem about Segments:***In any metric geometry, every**line segment*is a*convex set*.**Mark's Theorem about Points:***In any metric geometry, every**point*is a*convex set*.

For the first part of today's meeting, we will finish our discussion of *Convex Sets* with some **Presentations**.

**Nicole Adams and Michael Cooney and Jake Schneider: **(Study Book Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 pages 1 - 14) Frick, Frack, and Moonbeam came up with a theorem, and they each came up with their own proof.

**Frick's proof: **

- Suppose \( \Delta PQR\) is given.
- Then \( \Delta PQR = \overline{PQ} \cup \overline{QR} \cup \overline{RP} \) (by definition of
*triangle*). - Line segments are
*convex sets*. - The
*union*of*convex sets*will always be another*convex set*. - Therefore \( \Delta PQR\) is
*convex*.

**Nicole Adams Presentation #3: **What is wrong with Frick's "proof"?

**Frack's proof: **

Consider the triangle \( \Delta PQR\) shown. Observe that for any points \( A,B \in \Delta PQR\), the line segment \( \overline{AB} \subset \Delta ABC\)

**Michael Cooney Presentation #3: **What is wrong with Frack's "proof"? Explain.

**Moonbeam's proof: **

Consider the triangle \( \Delta PQR\) shown. Observe that for the two points \( A,B \in \Delta PQR\), the line segment \( \overline{AB} \subset \Delta ABC\)

**Jake Schneider Presentation #2: **What is wrong with Moonbeam's "proof"? Explain.

**Class: **What is wrong with the *Frick Frack Moonbeam "Theorem"*?

**The Plane Separation Axiom**

Now that we have discussed **Partitions of Sets** and **Convex Sets**, we are ready to understand the terminology used in the **Plane Separation Axiom**. The axiom is presented in two places:

Observe the that the **Plane Separation Axiom** guarantees the first type of *desirable plane separation behavior* that was presented on page 2 of the Notes for Video 4.1 . Note that the other types of *desirable plane separation behavior* are not mentioned in any way. It will turn out that if a *metric geometry* satisfies the *Plane Separation Axiom*, it can be proven in *theorems* that the other types of *desirable plane separation behavior* will be guaranteed, as well. (We will learn that in the coming week.)

In order to *use* the *Plane Separation Axiom (PSA)* it will be crucial to recognize that statements (ii) and (iii) of the *PSA* are *conditional statements*, and to recognize how one *uses conditional statements* in a proof. This is discussed on pages 15 - 24 of the Notes for Video 4.1.

**Josh** will present an example that will involve using statements (ii) and (iii) of the *PSA* in a proof.

**Josh Stookey (Presentation #2): **
(Study Textbook Section 4.1, watch
Video 4.1, and study pages 15 - 28 of the Notes for Video 4.1). Suppose that \(l\) and \(m\) are lines in a metric geometry that satisfies the *Plane Separation Axiom (PSA)*. Furthermore suppose that lines \(l\) and \(m\) intersect at point \(P\) and that \(A,B,C\) are points on line \(m\) such that \(A-P-B-C\)

- Make a drawing to illustrate the configuration of lines and points
- Prove that \(A\) and \(B\) are on
*opposite sides*of line \(l\). That is, prove that they are in*different half planes*of line \(l\). - Prove that \(B\) and \(C\) are on the
*same side*of line \(l\). That is, prove that they are in*the same half plane*of line \(l\).

**Fri Feb 25 Meeting Topics**

**Final Topic from Section 4.1: Proving Same Side / Opposite Side Statements**

**Mark B: **On Wednesday, we discussed the **Plane Separation Axiom**. The axiom is presented in two places:

On Wednesday, Josh presented proofs of a couple of Opposite Side / Same Side Statements:

**Josh's Theorem: ** Suppose that \(l\) and \(m\) are lines in a metric geometry that satisfies the *Plane Separation Axiom (PSA)*. Furthermore suppose that lines \(l\) and \(m\) intersect at point \(P\) and that \(A,B,C\) are points on line \(m\) such that \(A-P-B-C\)

- Make a drawing to illustrate the configuration of lines and points
- Prove that \(A\) and \(B\) are on
*opposite sides*of line \(l\). That is, prove that they are in*different half planes*of line \(l\). - Prove that \(B\) and \(C\) are on the
*same side*of line \(l\). That is, prove that they are in*the same half plane*of line \(l\).

Remember that Josh's proofs made use of *PSA (ii)* and *PSA (iii)*. Well, more precisely, Josh's proofs made use of the *contrapositives* of those statements.

**Josh's proof of statement (a), paraphrased:**

- Suppose the given stuff.
- Observe that points \(A,B\) are not on line \(l\). (We know that points \(A,P,B,C\) are all on line \(m\), and we know that lines \(l,m\) intersect at \(P\), and we know that distinct lines can only intersect once. So \(A,B\) are not on \(l\).)
- Observe that \(P\in \overline{AB}\). (Because we know that \(A-P-B\), the definition of line segment tells us that \(P\) is in the segment.)
- So line \(l\) intersects \(\overline{AB}\) at a point \(P\) that is between \(A\) and \(B\).
- Therefore, points \(A\) and \(B\) lie in
*different half planes*of line \(l\). (by (4) and*PSA (ii) contrapositive*)

**Josh's proof of statement (b), paraphrased:**

- Suppose the given stuff.
- Line \(l\) intersects line \(m\) at point \(P\) (that is given) and, therefore, cannot intersect line \(m\) at any other points (Theorem 2.1.6 tells us that two distinct lines cannot intersect more than once.)
- Observe that \(P\notin \overline{BC}\). (Because we know that \(P-B-C\), the definition of line segment tells us that \(P\) is not in the segment.)
- So line \(l\) does not intersect \(\overline{BC}\).
- Therefore, points \(B\) and \(C\) lie in
*the same half plane*of line \(l\). (by (4) and*PSA (iii) contrapositive*)

Observe that Josh's proof show that sometimes Opposite Side / Same Side statements can be proven by using *PSA (ii) contrapositive* or *PSA (iii) contrapositive*.

But sometimes, Opposite Side / Same Side statements are proved simply by using *PSA (i)*.

**Mandie** will give a presentation about that.

**Mandie Dicicco (Presentation #3) **(Read Textbook Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1) Prove **Theorem 4.1.3:**

**Section 4.2 The Plane Separation Axiom (PSA) in Our Familiar Metric Geometries**

**Mark B: **As mentioned on Monday and Wednesday, all four of the *metric geometries* that we have encountered so far (*Euclidean plane, Taxicab Plane, Max Plane, Poincaré plane*) do satisfy the *Plane Separation Axiom (PSA)*. Section 4.2 of the book is devoted to proving that they do. (The *Max plane* is not included in the discussion, but the proof that it satisfies *PSA* would be similar to the proofs that the *Taxicab plane* satisfies *PSA*.)

As is often the case with our book, some of the most difficult material has to do with proving that certain *analytic geometries* do satisfy certain *axioms*. The details of the proofs in Section 4.2 are messy, and are not so necessary for our course. In our course, we're most interested in understanding the *axioms* and using them to *prove theorems* about *axiomatic geometry*. We like knowing about the *models* (such as the *Euclidean plane, Taxicab Plane, Max Plane, Poincaré plane*) because they illuminate the significance of certain *axioms*. But if certain calculations involving a particular *model* are too cumbersome, we can sometimes just skip the details and study the conclusions, without diminishing too much our understanding of the meaning of the *axioms*.

However, it is important to *read* Section 4.2, and to understand the *overall structure* of what is presented. A rough summary of the content of Section 4.2 is:

- Define sets that will play the role of
*half planes*for the*Euclidean plane*. - Prove that those
*half planes*do satisfy the*Plane Separation Axiom*. - Define sets that will play the role of
*half planes*for the*Poincaré plane*. - Prove that those
*half planes*do satisfy the*Plane Separation Axiom*.

You might notice that the *Taxicab plane* and *Max plane* are not included in the discussion in Section 4.2.. The *Taxicab plane* is discussed in one of the exercises; the *Max plane* is not discussed, because it behaves so similarly to the *Taxicab plane*.

In the videos,

- The presentation of the
*half planes*is found on pages 6,7,8 of the Notes for Video 4.2. - The discussion of the proofs is found on pages 9,10,11 of the Notes for Video 4.2.

The definition of the *half planes* for *Euclidean Geometry* seems reasonable enough, but there is some subtlety. The video describes that the authors of the textbook (who almost never make mistakes) give a bad definition in the book.

But the definion of *half planes* for the *Poincaré plane* might seem a little surprising, so it is worth exploring a bit. **Joe** will present an example that may clarify the situation.

**Joe Durk (Presentation #3): Illustrating Concavity of Poincaré Half Planes** (Read the Textbook Section 4.2 and watch Video 4.2 and study pages 8 and 13 of the Notes for Video 4.2)

- Draw the
*Poincaré Type II line*\( \ _0L_4\). - Shade
*half plane*\( \ _0H_4^+\). Observe that it does not look*convex*, but*half-planes*are*supposed to be convex*. - Add to your drawing points \(A=(-4,3)\) and \(B=(4,3)\) and segment \(\overline{AB}\). Observe that line \( \ _0L_4\)
*does not*intersect segment \(\overline{AB}\). - Make another drawing of
*Poincaré Type II line*\( \ _0L_4\) and shade*half plane*\( \ _0H_4^+\). - Add to your drawing points \(C=(0,3)\) and \(B=(4,3)\) and segment \(\overline{CB}\). Observe that line \( \ _0L_4\)
*does*intersect segment \(\overline{BC}\).

**Section 4.3 Pasch Geometries**

**Mark B: **We have discussed the fact that all four of the *metric geometries* that we have encountered so far (*Euclidean plane, Taxicab Plane, Max Plane, Poincaré plane*) do satisfy the *Plane Separation Axiom (PSA)*. So one might think that we don't need to make the *PSA* an axiom. Maybe we could simply *prove in a theorem* that all *metric geometries* satisfy the *PSA*. It turns out that proving such a theorem would be *impossible*, because the statement is *not true*. That is, there are *metric geometries* that *do not* satisfy the *PSA*. One of those *metric geometries* is the *Missing Strip Plane*, which is presented on page 79 of the textbook and on pages 14 - 17 of the Notes for Video 4.3

Recall that at the beginning of Video 4.1, some common behaviors of drawings was presented. One of these was the following:

**Kelsie** will present an example showing how that kind of behavior *might not happen* in the *Missing Strip Plane*.

**Kelsie Flick (Presentation #3): An example of the Missing Strip Plane not having nice plane separation behavior**
(read pages 14 - 21 of the Notes for Video 4.3

- Plot \(A=(-1,0)\), \(B=(3,0)\), \(C=(3,8)\), and \(P=(2,1)\) in the
*Missing Strip Plane*. - Draw \(\Delta ABC\) and shade \( \text{int}(\Delta ABC\)).
- Draw \(\overrightarrow{BP}\).

Observe that \(P \in \text{int}(\Delta ABC)\) and yet ray \(\overrightarrow{BP}\) *does not intersect* side \(\overline{AC}\) of \( \Delta ABC\).

**Book Sections and Videos: More discussion of these three sections:**

- 4.1 The Plane Separation Axiom (Video 4.1) (Notes for Video 4.1)
- 4.2 PSA for the Euclidean and Poincaré Planes (Video 4.2) (Notes for Video 4.2)
- 4.3 Pasch Geometries (Video 4.3) (Notes for Video 4.3)

**Exercises:**

**Suggested Exercises For Week 7 and 8:**- 4.1 # 1, 2, 4, 5, 6, 8, 9, 10, 11, 13
- 4.2 # 1, 4
- 4.3 # 1, 2, 3, 4, 7

**Assigned Homework Due During Week 8: H06**due at the start of the Mon Feb 28 class meeting.**H06 Cover Sheet**

**Mon Feb 28 Meeting Topics**

We discussed the proof of this theorem in groups and then as a class:

**Theorem: **In a metric geometry that satisfies the *Plane Separation Axiom (PSA)*, if line \(l\) intersects sides \(\overline{AB}\) and \(\overline{AC}\) of \(\Delta ABC\) at points \(D\) and \(E\) such that \(A-D-B\) and \(A-E-C\), then line \(l\) will not intersect side \(\overline{BC}\).

**Homework H06** due at the start of the Mon Feb 28 class meeting. **H06 Cover Sheet**

**Quiz Q06** during last part of the Mon Feb 28 class meeting.

**Wed Mar 2 Meeting Topics**

We discussed the graded Homework H06 papers and discussed more topics from Sections 4.1, 4.2, 4.3.

**Exam X2 (Covering Chapter 3 and Sections 4.1, 4.2, 4.3) for the full duration of the Fri Mar 4 class meeting**

**No Calculators (They are not needed, anyway.)**

**No Phones**

**No Books or Notes**

**For Reference**,You may use your **Theorem List** that I handed out in class on Mon, Feb 28.

- You may add to your Theorem List in advance of the Exam. (You may add whatever you can
*hand-write*on those pages; no additional pages.) - You may not share your Theorem List during the Exam.
- If you have lost your Theorem List, you may print one out from the above link. I will not have any available on the day of the exam.

**The Exam** is 5 problems, 4 pages, printed on front & back of two sheets of paper.

- Question about a
*Euclidean*or*Poincaré*object of one of these types:*segment*,*ray*,*angle**triangle*. (concepts from Section 3.4) - Question about
*Convex Sets*(concepts from Section 4.1) - Question about
*Rulers*(concepts from Section 3.2 and 3.3) - Question about
*Betweenness*(concepts from Section 3.2) - Question involving
*Plane Separation*(concepts from Sections 4.1 and 4.3)

**Sat Mar 5 - Sun Mar 13 is Spring Break.**

**Book Sections and Videos:**

- 4.4 Interiors and the Crossbar Theorem (Video 4.4) (Notes for Video 4.4)

**Exercises:**

**Suggested Exercises:**- 4.4 # 2,3,4,5,6,9,10,11,12,13,15,17,19,23,25

**Assigned Homework H07**due at the start of the Fri Mar 18 class meeting.**H07 Cover Sheet**

**Mon Mar 14 Meeting Topics**

The textbook does not define the **interior a segment** and **interior of a ray** until the current Section 4.4. But there is nothing in the definition of **interior of a segment** or **interior of a ray** that needed to wait until now to be introduced. Everything in the definitions could have been introduced back in **Section 3.3**, when **segment** and **ray** were defined. In fact, the *interiors* of *segments* and *rays*
*were* introduced in Video 3.3a. Observe that the *interior* of a *segment* or *ray* is the same thing as the set of *passing points* for that object. (*Passing points*
were also defined in Video 3.3a.)

The fact that *lines*, *segments*, and *rays* are *convex* was discussed in a meeting back when we were discussion **Section 3.3**.

- In a Class Presentation, Howard proved that a
*line*is a convex set. - In another Class Presentation, Jingmin showed drawings that indicated that the proof that a
*ray*is a convex set would involve many*cases*. But*conceptually*, each case would be no harder than the proof that Howard did for*lines*.

In the current **Section 4.4**, it is discussed that *interiors* of *segments* and *rays* are also convex. The *proof* of the fact is left as an exercise. We won't discuss the proof, and the exercise is not assigned, but we will *use* that fact that *interiors* of *segments*, and *rays* are convex, even though we haven't done the proof. It will be helpful to keep in mind a
visualization of those facts from Video 4.4.

A fairly unsurprising and uninteresting Theorem 4.4.2 is presented in the book and in Video 4.4. Even though this theorem is not surprising, it is in fact very important, and gets used throughout the rest of Chapter 4 and occasionally throughout the rest of the book.

In fact, Theorem 4.4.2 is used right away in the book, to prove Theorem 4.4.3 (the Z Theorem). This theorem might also look uninteresting, but it will be key to proving the **Crossbar Theorem**, later in the section.

In Video 4.4, more descriptive half-plane notation is introduced. This notation is not used in the book, but it is easy to understand and is very helpful. Later in Video 4.4, the new notation is used in a definition of *interiors of angles and triangles*.

The textbook presents a theorem about angle interiors that is really just a *restating of the definition*, using the terminology of *same side* instead of the terminology of *half planes*:

Earlier in the meeting, it was mentioned that although Theorem 4.4.2 may look uninteresting, it is very important. We have already seen that **Theorem 4.4.2** is used in Proving **Theorem 4.4.3 (the Z Theorem)**. The Theorem 4.4.2 is also used in proving the Theorem 4.4.6, a theorem that is easy to visualize and very useful.

**Remark: **Notice how the *Statement of Theorem 4.4.6* is *illustrated* in the video: There is one drawing for the situation described in the *hypothesis*, and another drawing for the statement described in the *conclusion*.

In your **Assigned Homework H07**, you will be asked to ** Justify** and

Similar to the statement of **Theorem 4.4.6**, just discussed, is the following

You are asked to ** Prove** the statement in your

Remember that the *interior of an angle* is defined to be an intersection of *half planes*. When one wants to prove that a particular point either *is* or *is not* in the interior of some angle, it will often be useful to invoke the
Plane Separation Axiom Parts (ii) and (iii) and their contrapositives, which describe how half planes behave. An example of a problem involving this very important technique is the following:

**Rachel** will illustrate the statement with a drawing.

**Rachel Han (Presentation #3): Rewriting and Illustrating a Statement about Angle Interiors**
(read pages 16 - 17 of the Notes for Video 4.4)
Use a drawing to illustrate the statement of

**Mark B** will prove the statement that Rachel just illustrated. Notice how the proof uses the *Plane Separation Axiom Parts (ii) and (iii) and their contrapositives*.

A very famous Theorem in Axiomatic Geometry is the Crossbar Theorem. Its proof is too complex to discuss in detail in class. Read the book's proof, or study the proof in the video. (The proof in the video just expands, justifies, and illustrates the book's proof.)

In the book, the **Crossbar Theorem** is *immediately* followed by two more theorems whose proofs use the Crossbar Theorem. There are many interesting and useful *Theorems* and *Statements* presented in Section 4.4, both in the *Exposition* and in the *Exercises*. Ideally, one would have time to learn to *prove* all of them. Most of you probably do not have that much time to spend on such a project. However, you should at least take the time to do two things:

*Read*the Theorems and Statements, and make*drawings*to illustrate them, to help you fully understand what they*mean*.- Once you have made a
*drawing*to illustrate a Theorem or Statement, think a bit about how its*proof*might be*structured*.

**Kyle Hill (Presentation #3):**
(read pages 16 - 17 of the Notes for Video 4.4) Make a drawing to illustrate the statement of this theorem:

**Jack Lazenby (Presentation #3):**
(read pages 16 - 17 of the Notes for Video 4.4) Make a drawing to illustrate the statement of this theorem:

**Group Activity: Illustrating a Statement about Angle Interiors**
Make a drawing to illustrate this Statement:

The two **Presentations** and **Group Activity** just finished were about *visualizing* statements about angle interiors. Two of those were *If and only If* statements. When considering how to structure the proof of an *If and Only If* statement, it may be helpful to first *re-write* the statement as an *Equivalence* statement. Then, it will become clearer that a *two-part proof* will be necessary. **James** will do some of that *rewriting* and *thinking about proof structure* in his **Presentation**.

**James Foley (Presentation #3):** Earlier, **Kyle** and **Jack** illustrated the statements of **Theorem 4.4.8** and **Theorem 4.4.9**. Both of those Theorems are presented in the book as *If and Only If* statements.

Your job is to do two things for each theorem:

- Rewrite the statement of the theorem as an
*Equivalence*statement. - Write the
*frame*of the proof, with the major parts of the proof indicated with their objectives, and the first and last statements of each part of the proof provided. Show the*gap*between the first and last statements clearly.

**Mark B** will *prove* the two theorems. Notice that the proof of **Theorem 4.4.9** is cool because it can be significantly *shortened* by doing a clever *change of variables*.

**Wed Mar 16 Meeting Topics**

The *Converse of the Statement of The Crossbar Theorem* is presented as a *Theorem* in Video 4.4. The statement is *not* presented as a Theorem in the book. Rather, it is just presented as a *Statement to be Proven* in **Book Exercise 4.4#12**. You will prove the statement in your **Homework H07 Problem [5]**.

Here is a nice Example that uses both the **Crossbar Theorem** and the **Converse of the Statement of the Crossbar Theorem**.

If \(\angle ABC = \angle DBE\) and ray \(\overrightarrow{BF}\) intersects \(\text{int}(\overline{AC})\),

then ray \(\overrightarrow{BF}\) also intersects \(\text{int}(\overline{DE})\).

**Suggested Exercise 4.4#17** says to prove the following:

then \( \ l\cap\Delta ABC \ \) has exactly two points.

A slogan that conveys some elements of that statement could be:

This statement is actually fairly important, and it is rather difficult to prove. It really ought to be called a *Theorem*. Mark B will describe the main elements of the proof.

Having seen that

a natural question is,

**Exercise 4.4#23** Deals with this question. That exercise says,

An informal wording of this question could be:

If we only read the statement of the exercise, we might be duped into thinking that the answer to our question is, *an angle cannot enclose a line*.

And if we draw an example of an *angle* and a *line* that *passes through a point in the interior of the angle*, it seems that the line will always intersect the angle at one or two points. That is, the drawing seems to confirm that *an angle cannot enclose a line*.

But realize that **Exercise 4.4#23** is another one of those * Part B* exercises with the instructions,

In other words, the *full* instructions for **Exercise 4.4#23** are,

That is,

If we draw angles in the *Euclidean Plane* (Mark B will do this.), it sure seems like *an angle cannot enclose a line*. That is, it seems lke the statement is *true*.

But the fact that we are asked to **prove or disprove** the statement means that we should not so blithely assume that the statement is *true*, even if drawings seem to indicate that the statement *is* true.

**Taylor** and **Jack** have **Presentations** that will shed light on the subject.

**Taylor Miller (Presentation #3): **

- Make a big, clear drawing of
*Poincaré*angle \(\angle ABC\) for \(A=(-4,5)\), \(B=(0,3)\), and \(C=(4,5)\). - On your drawing, shade \(\text{int}(\angle ABC)\).
- Add
*Poincaré*line\( \ _0L_{10}\) to your drawing.

**Jack Muslovski (Presentation #3): **

- Make a big, clear drawing of
*Poincaré*angle \(\angle ABC\) for \(A=(0,8)\), \(B=(0,5)\), and \(C=(4,3)\). - On your drawing, shade \(\text{int}(\angle ABC)\).
- Add
*Poincaré*line\( \ _7L\) to your drawing.

**Definition of Crossbar Interior**

**Symbol:**\( \ \text{cint}(\angle ABC) \)**Words:***the***crossbar interior of \( \ \angle ABC \ \)****Usage:**\( \ \angle ABC \ \) is an angle in a*metric geometry*.**Meaning:**The set \( \{ P | D-P-E \ \text{ for some } D\in \text{int} (\overrightarrow{BA}) \text{ and some } E \in \text{int} (\overrightarrow{BC}) \} \)

Observe that the definition of the **crossbar interior of \( \ \angle ABC \ \)** only uses the concept of **betweenness**. So \( \ \text{cint}(\angle ABC) \) is defined for any angle in any **metric geometry**.

Now recall that the **interior of \( \ \angle ABC \ \)**, introduced in Section 4.4, uses the concept of **half planes**. So \( \ \text{int}(\angle ABC) \) is only defined for angles in a **metric geometry that has half planes**. That is, \( \ \text{int}(\angle ABC) \) is only defined for angles in a **Pasch geometry**.

A natural question is,

Abbreviated in symbols,

**Exercise 4.4#25** Deals with this question. That exercise says,

If we only read the statement of the exercise, we might be duped into thinking that the answer to our question is, *yes, the crossbar interior of the angle is the same as the interior of the angle*.

But realize that **Exercise 4.4#25** is one of those * Part B* exercises with the instructions,

In other words, the *full* instructions for **Exercise 4.4#25** are,

**Logan** will shed some light on this question in his **Presentation**.

**Logan Prater (Presentation #3): **

- Make a big, clear drawing of
*Poincaré*angle \(\angle ABC\) for \(A=(0,8)\), \(B=(0,5)\), and \(C=(4,3)\). - On your drawing, shade \(\text{int}(\angle ABC)\).
- Add
*Poincaré*line\( \ _7L\) to your drawing.

**Fri Mar 18 Meeting Topics**

**Assigned Homework H07** due at the start of the Fri Mar 18 class meeting. **H07 Cover Sheet**

**Quiz Q07** during last part of the Fri Mar 18 class meeting

**Book Sections and Videos:**

- Section 5.1 The Measure of an Angle (Video 5.1) (Notes for Video 5.1)
- Section 5.3 Perpendicularity and Angle Congruence

**Suggested Exercises:**

- 5.1 # 1, 2, 3, 7
- 5.3 # 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 18, 19, 20, 21

**Assigned Homework H08** due at the start of the Fri Mar 25 class meeting. **H08 Cover Sheet**

**Mon Mar 21 Meeting Topics**

**Sammi Rinicella (Presentation #3): **
To prepare for your presentation, read page 2 of the Notes for Video 5.1.

Draw points \(A=(1,0)\), \(B=(0,0)\), \(C=(2,0)\), \(D=(-3,0)\). (Make a big, clear drawing.)

- In a
*high school*geometry course, what would be the*measure*of angle \( \angle ABC \)? - There are two reasons that the answer to (a) cannot work for us in
*MATH 3110*. One has to do with the*definition*of*angle*, and the other has to do with statement*(i)*(that is,*Axiom (i)*) in the*definition*of*angle measure*. Explain. - In a
*high school*geometry course, what would be the*measure*of angle \( \angle ABD \)? - There are two reasons that the answer to (c) cannot work for us in
*MATH 3110*. One has to do with the*definition*of*angle*, and the other has to do with the*definition*of*angle measure*. Explain.

On **page 84 of the book**, there are a few paragraphs that are meant to walk the reader through the process of figuring out what the formula for the **Euclidean Tangent to a Poincaré Ray \(\overrightarrow{PQ}\)** should be. Near the middle of these paragraphs, the authors say,

The reader might not find those paragraphs of explanation very helpful: The expression \( \pm(y_B,c-x_B)\) is kind of pulled out of thin air.

Following these paragraphs, the **Definition of the Euclidean Tangent to a Poincaré Ray \(\overrightarrow{PQ}\)** is presented, and the definition incorporates the expressions \( (y_B,c-x_B)\) and \( -(y_B,c-x_B)\).

On **Page 16 of the Notes for Video 5.1**, the same **Definition of the Euclidean Tangent to a Poincaré Ray** is presented, although with letters changed, and there is also a **Procedure for Computing \(T_{PQ}\), the Euclidean Tangent to a Poincaré Ray \(\overrightarrow{PQ}\)**. There is no explanation at all of *why* the formulas look the way they do.

**Jake, Helen, and Danielle** will do **Presentations** that may help you better understand why the formulas are what they are.

**Jake Schneider (Presentation #3): **
To prepare for your presentation, read pages 15 - 17 of the Notes for Video 5.1.

- Make a large, clear drawing of Poincaré Ray \(\overrightarrow{BA}\), with \(B=(3,4)\) and \(A=(2,3)\). Be sure to label stuff clearly:
- Draw the whole
*Type II*line \(\overleftrightarrow{AB}\), with the part of the line that is \(\overrightarrow{BA}\) drawn solid and bold, and the part of the line that is not \(\overrightarrow{BA}\) drawn dotted. Label the line with its \( \ _cL_r\) description. - Label important locations with their \((x,y)\) coordinates: The points \(A\) and \(B\). The center of the circle \((c,0)\). The missing endpoints \((c-r,0)\) and \((c+r,0)\).

- Draw the whole
- Add to your drawing a vector with tail at \(B\) and head at the center of the circle \((c,0)\). (The vector should be a regular,
*straight-looking*,*(Euclidean)*vector.) Put the \((x,y)\)*vector components*of this vector alongside the vector. - Add to your drawing a new vector that results from
*rotating*the vector from part (b)*clockwise*\(90^\circ\) around point \(B\). (The rotated vector should also be a regular,*straight-looking*,*(Euclidean)*vector.) Put the \((x,y)\)*vector components*of this new vector alongside the vector. - Compute the components of \(T_{BA}\) using the
**Procedure for Computing \(T_{PQ}\)**on Page 16 of the Notes for Video 5.1 - How do your results from (c) and (d) compare?

**Helen Sitko (Presentation #3): **
To prepare for your presentation, read pages 15 - 17 of the Notes for Video 5.1.

- Make a large, clear drawing of Poincaré Ray \(\overrightarrow{BC}\), with \(B=(3,4)\) and \(C=(6,5)\). Be sure to label stuff clearly:
- Draw the whole
*Type II*line \(\overleftrightarrow{BC}\), with the part of the line that is \(\overrightarrow{BC}\) drawn solid and bold, and the part of the line that is not \(\overrightarrow{BC}\) drawn dotted. Label the line with its \( \ _cL_r\) description. - Label important locations with their \((x,y)\) coordinates: The points \(B\) and \(C\). The center of the circle \((c,0)\). The missing endpoints \((c-r,0)\) and \((c+r,0)\).

- Draw the whole
- Add to your drawing a vector with tail at \(B\) and head at the center of the circle \((c,0)\). (The vector should be a regular,
*straight-looking*,*(Euclidean)*vector.)Put the \((x,y)\)*vector components*of this vector alongside the vector. - Add to your drawing a new vector that results from
*rotating*the vector from part (b)*counterclockwise*\(90^\circ\) around point \(B\). (The rotated vector should also be a regular,*straight-looking*,*(Euclidean)*vector.) Put the \((x,y)\)*vector components*of this new vector alongside the vector. - Compute the components of \(T_{BC}\) using the
**Procedure for Computing \(T_{PQ}\)**on Page 16 of the Notes for Video 5.1 - How do your results from (c) and (d) compare?

**Danielle Stevens (Presentation #3): **
To prepare for your presentation, read pages 15 - 17 of the Notes for Video 5.1.

- Make a large, clear drawing of Poincaré Ray \(\overrightarrow{CB}\), with \(B=(3,4)\) and \(C=(6,5)\). Be sure to label stuff clearly:
- Draw the whole
*Type II*line \(\overleftrightarrow{PBC}\), with the part of the line that is \(\overrightarrow{CB}\) drawn solid and bold, and the part of the line that is not \(\overrightarrow{CB}\) drawn dotted. Label the line with its \( \ _cL_r\) description. - Label important locations with their \((x,y)\) coordinates: The points \(B\) and \(C\). The center of the circle \((c,0)\). The missing endpoints \((c-r,0)\) and \((c+r,0)\).

- Draw the whole
- Add to your drawing a vector with tail at \(C\) and head at the center of the circle \((c,0)\). (The vector should be a regular,
*straight-looking*,*(Euclidean)*vector.)Put the \((x,y)\)*vector components*of this vector alongside the vector. - Add to your drawing a new vector that results from
*rotating*the vector from part (b)*clockwise*\(90^\circ\) around point \(C\). (The rotated vector should also be a regular,*straight-looking*,*(Euclidean)*vector.) Put the \((x,y)\)*vector components*of this new vector alongside the vector. - Compute the components of \(T_{CB}\) using the
**Procedure for Computing \(T_{PQ}\)**on Page 16 of the Notes for Video 5.1 - How do your results from (c) and (d) compare?

**Wed Mar 23 Meeting Topics**

**Josh Stookey Presentation #3: **
(Read Section 5.1, watch Video 5.1, and study the Notes for Video 5.1)
Let \(A=(3,4), B=(9,4), C=(15,4)\)

- If
*Euclidean*angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_E(\angle ABC)\). If*Euclidean*angle \(\angle ABC\) does not exist, explain*why*it does not exist. - If
*Poincaré*angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_H(\angle ABC)\). If*Poincaré*angle \(\angle ABC\) does not exist, explain*why*it does not exist.

**Hannah Worthington Presentation #3: **
(Read Section 5.1, watch Video 5.1, and study the Notes for Video 5.1)
Let \(A=(2,3), B=(6,5), C=(9,4)\)

- If
*Euclidean*angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_E(\angle ABC)\). If*Euclidean*angle \(\angle ABC\) does not exist, explain*why*it does not exist. - If
*Poincaré*angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_H(\angle ABC)\). If*Poincaré*angle \(\angle ABC\) does not exist, explain*why*it does not exist.

**Aoife Zoerche Presentation #3: **
(Read Section 5.1, watch Video 5.1, and study the Notes for Video 5.1)
Let \(A=(3,7), B=(3,4), C=(10,3)\).

- Make a large, clear drawing of
*Poincaré angle*\(\angle ABC\). Be sure to label stuff clearly:- Draw whole
*Poincaré lines*, with the parts of lines that are*rays*of the*Poincaré angle*\(\angle ABC\) drawn solid and bold, and the parts of lines that are not part of those rays drawn dotted. Label all*Poincaré lines*with their \( \ _aL\) or \( \ _cL_r\) description. - Label important locations with their \((x,y)\) coordinates: The points \(A,B,C\). The
*missing endpoint*of every*Type I*line. The*missing endpoints*and*center*of every*Type II*line.

- Draw whole
- Add to your drawing the
*Euclidean tangent to Poincaré ray*\(\overrightarrow{BA}\) and the*Euclidean tangent to Poincaré ray*\(\overrightarrow{BC}\), and label them \(T_{BA}\) and \(T_{BC}\). - Add the \((x,y)\)
*vector components*to those two tangent vectors. - Find \(m_H(\angle ABC)\). Show details of the computation clearly, and give an exact, simplified answer and a decimal approximation.

**Fri Mar 25 Meeting Topics**

**Howard Bartels (Presentation #3): ** Prove the following Theorem

**Jingmin Gao (Presentation #3): **The book presents the definition of a **vertical pair** of angles on page 104. The illustration that is shown is of a **vertical pair** in the *Euclidean plane*. (Howard probably used a similar illustration in his presentation.)

Draw an example of a **vertical pair** in the *Poincaré Upper Half Plane*.

**Nicole Adams (Presentation #4):**

- In the
*Euclidean plane*, let $$A=(1,0), \ \ B=(3,0), \ \ C=(0,1), \ \ D=(2,1), \ \ M=(2,0) \ \ O=(0,0)$$ Make a big, clear drawing showing the following objects in the*Euclidean Plane*.- Points \(A,B,C,D,M\)
- Segment \(\overline{AB}\)
- Line \(\overleftrightarrow{CM}\)
- Line \(\overleftrightarrow{DM}\)
- Line \(\overleftrightarrow{CO}\)

- Answer the following questions, using your drawing as an illustration:
- Questions about
*bisector*of a*segment*- What is a
*bisector*of a*segment*? (Explain) - Does every
*segment*have a*bisector*? (Explain) - Can a
*segment*have*more than one bisector*? (Explain)

- What is a
- Questions about a
*perpendicular to a segment*- What is a
*perpendicular to a segment*? (Explain) - Does every
*segment*have a*perpendicular*? (Explain) - Can a
*segment*have*more than one perpendicular*? (Explain)

- What is a
- Questions about a
*perpendicular bisector*of a*segment*- What is a
*perpendicular bisector*of a*segment*? (Explain) - Does every
*segment*have a*perpendicular bisector*? (Explain) - Can a
*segment*have*more than one perpendicular bisector*? (Explain) - Is every
*bisector*of a*segment*a*perpendicular bisector of the segment*? - Is every
*perpendicular to a segment*a*perpendicular bisector of the segment*?

- What is a

- Questions about

**Homework H08** due at the start of the Mar 25 class meeting. **H08 Cover Sheet**

**Quiz Q08** will be a **take-home quiz**, due **Mon Mar 28**.

**Book Sections and Videos:**

- Section 4.5 (for Definition of Quadrilateral)
- Section 6.1 The Side-Angle-Side Axiom
- Section 6.2 Basic Triangle Congruence Theorems (Video 6.2) (Notes for Video 6.2)

**Suggested Exercises:**

- 6.1 # 1, 2, 4, 5, 6, 7, 8, 9, 10, 12
- 6.2 # 1, 2, 3, 5, 6, 7, 8, 9, 12, 13

**Assigned Homework H09** due at the start of the Fri Apr 1 class meeting. **H09 Cover Sheet**

**Mon Mar 28 Meeting Topics**

**Wed Mar 30 Meeting Topics**

**Michael Cooney (Presentation #4): **Illustrate the statement that is to be proven in book exercise **6.1#10** on page 130. (**Don't prove the statement!**)

**Mandie Dicicco (Presentation #4): **Illustrate the statement that is to be proven in book exercise **6.2#5** on page 134. (**Don't prove the statement!**)

**Joe Durk (Presentation #4): **Illustrate the statement that is to be proven in book exercise **6.2#6** on page 134. (**Don't prove the statement!**)

**Kelsie Flick (Presentation #4): **Prove that ** Triangle Congruence** is

**James Foley (Presentation #4): **Prove that ** Triangle Congruence** is

**Fri Apr 1 Meeting Topics**

**Assigned Homework H09** due at the start of the Fri Apr 1 class meeting. **H09 Cover Sheet**

**Quiz Q09** Take-Home Quiz, will be distributed in class on Fri Apr 1; due at the start of the Mon Apr 4 class meeting

**Book Sections and Videos:**

- Section 6.2 Basic Triangle Congruence Theorems (Video 6.2) (Notes for Video 6.2)
- Section 6.3 The Exterior Angle Theorem and Its Consequences (Videos forthcoming)
- (Video 6.3a) (Notes for Video 6.3a)
- (Video 6.3b) (Notes for Video 6.3b)
- Section 6.4 Right Triangles

**Suggested Exercises:**

- 6.2 # 1, 2, 3, 5, 6, 7, 8, 12, 15
- 6.3 # 1, 2, 3, 4, 5, ,8, 10
- 6.4 # 1, 2, 3a, 6, 8, 10, 12

**Assigned Homework H10** due at the start of the Fri Apr 8 class meeting. **H10 Cover Sheet**

**Mon Apr 4 Meeting Topics**

**Quiz Q09** Take-Home Quiz, will be distributed in class on Fri Apr 1; due at the start of the Mon Apr 4 class meeting

**Wed Apr 6 Meeting Topics**

**Different versions of the Statement of Theorem 6.3.6**

**(Without Naming Vertices) Theorem 6.3.6:***In a Neutral Geometry triangle, if one side is bigger than another side,*

then the angle opposite the bigger side is longer than the angle opposite the smaller side.**(Abbreviated, Informal version) Theorem 6.3.6:***In a Neutral Geometry triangle, if BS then BA*.**(Abbreviated, Informal version with symbols) Theorem 6.3.6:***In a Neutral Geometry triangle, \( BS \rightarrow BA \)*.**(With Vertices Named) Theorem 6.3.6:***In every Neutral Geometry triangle \(\Delta ABC \), if \( AB > AC \) then \( m(\angle ACB) > m(\angle ABC)\).*

**Mandie Dicicco (Presentation #4):** (on the chalkboard) Illustrate the **Statement of Theorem 6.3.6**. (Illustrate the version of the theorem with the vertices named \(A,B,C\).) Draw the triangles with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.

**Proof of Theorem 6.3.6:**

- Suppose that in a neutral geometry, \(\Delta ABC\) has \(AB > AC\).
- There exists a point \(G \in int(\overline{AB}) \) such that \(\overline{AG} \simeq \overline{AC}\)
- \(G \in int(\angle ACB \)
- \( m(\angle ACB) > m(\angle ACG)\)
- \( m(\angle ACG) = m(\angle AGC)\)
- \( m(\angle AGC) > m(\angle ABC)\)
- \( m(\angle ACB) > m(\angle BC)\)

**End of Proof**

I've assigned the job of illustrating and justifying the statements in the proof to seven of you, as follows:

**Rachel Han (Presentation #4):**(on the chalkboard) Illustrate**Statement 1**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.**Kyle Hill (Presentation #4):**(on the chalkboard) Illustrate and Justify**Statement 2**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. (**Hint:**Use a theorem from Section 6.3.**Jack Lazenby (Presentation #4):**(on the chalkboard) Illustrate and Justify**Statement 3**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. (**Hint:**Use a Theorem from Section 4.4.)**Taylor Miller (Presentation #4):**(on the chalkboard) Illustrate and Justify**Statement 4**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. (**Hint:**Use the*Angle Measurement Axioms*(the*Definition of Angle Measure*) from Ch. 5.)**Jack Muslovski (Presentation #4):**(on the chalkboard) Illustrate and Justify**Statement 5**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. (**Hint:**Use a theorem from Section 6.1.)**Logan Prater (Presentation #4):**(on the chalkboard) Illustrate and Justify**Statement 6**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. (**Hint:**Use a theorem from Section 6.3.)**Sammi Rinicella (Presentation #4):**(on the chalkboard) Illustrate and Justify**Statement 7**of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.

**Jake Schneider (Presentation #4):**

- (on the chalkboard) Write the
of the statement of Theorem 6.3.6.*contrapositive* - Is the
*contrapositive*a true statement? Explain.

**Different versions of the Statement of Theorem 6.3.6**

**(Without Naming Vertices) Theorem 6.3.7:***In a Neutral Geometry triangle, if one angle is bigger than another angle,*

then the side opposite the bigger angle is longer than the side opposite the smaller angle.**(Abbreviated, Informal version) Theorem 6.3.7:***In a Neutral Geometry triangle, if BA then BS*.**(Abbreviated, Informal version with symbols) Theorem 6.3.7:***In a Neutral Geometry triangle, \( BA \rightarrow BS \)*.**(With Vertices Named) Theorem 6.3.7:***In every Neutral Geometry triangle \(\Delta ABC \), if \( m(\angle ACB) > m(\angle ABC) \) then \( AB > AC \).*

**Helen Sitko (Presentation #4):** (on the chalkboard) Illustrate the **Statement of Theorem 6.3.7**.
Draw the triangles with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.

**Danielle Stevens (Presentation #4):**

- (on the chalkboard) Write the
of the statement of Theorem 6.3.7.*contrapositive* - Is the
*contrapositive*a true statement? Explain.

**Fri Apr 8 Meeting Topics**

**Definition of Right Triangle** and associated other objects, from page 143 of the book.

Discussed **Theorem 6.4.1** (really a *Corollary* of previously-discussed theorems) *In a neutral geometry, there is only one right angle and one hypotenuse for each right triangle. The remaining angles are acute, and the hypotenuse is the longest side of the triangle.*

Discussed the **Theorem 6.4.2 (Perpendicular Distance Theorem): **Given a line \(l\) and a point \(P\) not on \(l\), among all the points \(Q\) on \(l\), the point \(Q \in l\) for which the segment \(\overline{PQ}\) is the *shortest* is the unique point \(Q \in l\) such that \(\overleftrightarrow{PQ} \perp l\). This enables us to define the ** distance from \(P\) to \(l\)** to be the length of that shortest segment.

Discussed ** Altitude lines** and

**Assigned Homework H10** due at the start of the Fri Apr 8 class meeting. **H10 Cover Sheet**

**Book Sections and Videos:**

- Section 6.4 Right Triangles
- Section 6.5 Circles and their Tangent Lines

**Suggested Exercises:**

- 6.4 # 1, 2, 3a, 6, 8, 10, 12
- 6.5 # 3, 9, 11, 14, 15, 16, 21, 26 (
**Note:**The statements in problems 14, 15, 16, 21, 28 are all*true*, so you should be trying to*prove*, not trying to*disprove them*.)

**Mon Apr 11 Meeting Topics**

Discussed **Theorem 6.4.4 (Hypotenuse-Leg HL): **

**Corollary 5.3.7**, from the previous chapter, told us that in a protractor geometry, every line segment has a unique ** perpendicular bisector**.

**Theorem 6.4.6** articulates an interesting fact about perpendicular bisectors, the theorem says that the following two statements are equivalent:

- Point \(P\) lies on the
*perpendicular bisector*of a line segment \(\overline{AB}\). That is, \(P\in l\) where \(l\) is the perpendicular bisector of \(\overline{AB}\). - Point \(P\) is
*equidistant*from the two endpoints of the line segment. That is, \(PA=PB\).

**Theorem 5.3.8**, from the previous chapter, told us that in a protractor geometry, every angle has a unique ** angle bisector**.

**Theorem 6.4.7** and **Exercise 6.4#11** articulate an interestings fact about angle bisectors.

**Theorem 6.4.7** says that if \(\overrightarrow{BD}\) is the bisector of \(\angle ABC\), then point \(D\) is *equidistant* from the two rays of the angle. That is, \(d(D,\overleftrightarrow{BA})=d(D,\overleftrightarrow{BC})\).

**Exercise 6.4#11** says that a *converse* statement is also true. A simplified version of the statement is the following:

*If point \(D \in \text{int}(\angle ABC)\) and \(D\) is equidistant from the two rays of the angle. (That is, \(d(D,\overleftrightarrow{BA})=d(D,\overleftrightarrow{BC})\).)
then \(\overrightarrow{BD}\) is the bisector of \(\angle ABC\).*

Realize that the results of **Theorem 6.4.7** and **Exercise 6.4#11** could be generalized and combined into a single *corollary* about *equivalent statements*.

**Unstated Corollary of Theorem 6.4.7 and Exercise 6.4#11** In a neutral geometry, the following two statements are equivalent:

- Point \(D\) lies on the
*bisector*of angle \(\angle ABC\). - Point \(D\) is the same point as point \(B\), or \(D \in \text{int}(\angle ABC)\) and \(D\) is
*equidistant*from the two rays of the angle. (That is, \(d(D,\overleftrightarrow{BA})=d(D,\overleftrightarrow{BC})\).

- \(\mathscr{C}_r(C)\), the
*Circle with Center \(C\) and Radius \(r\)* *Chord**Diameter Segment*(book calls this a*diameter**Interior of the Circle**Exterior*

**Exam X3 for the full duration of the Wed Apr 13 class meeting**

Five Problems:

- One Problem from Section 4.4 Interiors and the Crossbar Theorem
- One Problem from Section 5.3 Perpendicularity and Angle Congruence
- One Problem from Section 6.2 Basic Triangle Congruence Theorems
- One Problem from Section 6.3 The Exterior Angle Theorem and Its Consequences
- One Problem from Section 6.4 Right Triangles

**Reference for use during the exam: ****Axioms, Definitions, and Theorems Through Chapter 6**

**Fri Apr 15 Meeting Topics**

**[Example 1]: **Circle centered at \((C=0,0)\) with radius \(r=1\) in the *Euclidean Plane*

**[Example 2]: **Circle centered at \((C=0,0)\) with radius \(r=1\) in the *Taxicab Plane*

**[Example 3]: **The set of points

**Review Theorem 6.4.6** (about Perpendicular Bisectors of Line Segments) discussed on Monday.

**Associated Fact about Circles: Corollary 6.5.4 about Perpendicular Bisectors of Chords: ***For any circle in a neutral geometry, the perpendicular bisector of any chord contains the center of the circle.*

**Remark: **In the book, **Corollary 6.5.4** comes *after* **Theorem 6.5.3**, which the reader might take to mean that the Corollary follows from the Theorem 6.5.3 just presented. This is misleading. The corollary is a corollary of the *earlier* Theorem 6.4.6.

**Corollary of that Corollary: ***For any circle in a neutral geometry, the perpendicular bisectors of any two chords intersect at the center of the circle. (They may also be the same line, in which case they intersect at lots of points!)*

**Another Fact about Circles: Theorem 6.5.3***In any Neutral Geometry, if two circles have three points in common, then they are the same circle.*

*Interior of the Circle**Interior of the Circle**Tangent to the Circle**Secant of the Circle*

**Theorem 6.5.6: **In a Neutral Geometry, a line that intersects a circle \( \mathscr{C}_r(C) \) at a point \(Q\) is tangent to a circle if and only if the line is perpendicular to the radius segment \(\overline{CQ}\)

**Book Sections and Videos:**

- Section 6.5 Circles and their Tangent Lines
- Section 7.1 The Existence of Parallel Lines
- Portions of Section 7.2 Saccheri Quadrilaterals
- Portions of Section 7.3 The Critical Function

**Suggested Exercises:**

- 6.5 # 3, 9, 11, 14, 15, 16, 21, 26 (
**Note:**The statements in problems 14, 15, 16, 21, 28 are all*true*, so you should be trying to*prove*, not trying to*disprove them*.)

**Mon Apr 18 Meeting Topics**

**Wed Apr 20 Meeting Topics**

**Theorem 6.5.3** says that in *Neutral Geometry*, if two circles have three points in common, then the circles are in fact the same circle.

In the proof of that theorem, there is a little *mini-proof* of the following **Fact** that is key to the larger proof:

**Fact Proven Inside the Proof of Theorem 6.5.3: **In *Neutral Geometry* if distinct points \(R,S,T\) lie on a common circle, then the *perpendicular bisectors* of segments \(\overline{RS}\) and \(\overline{ST}\) intersect.

**Nicole Adams (Presentation #5): **Write the *contrapositive* of the **Fact** just stated.

**Kelsie Flick (Presentation #5)(Suggested Exercise 6.5#26): **Give an example of three

(

(E-mail Mark for help if you need it.)

**Josh Stookey (Presentation #4): **In *neutral geometry*, given

- a circle \(\mathscr{C}_r(C)\)
- a chord \(\overline{AB}\) that is
*not*a*diameter* - a point \(D\) such that \(A-D-B\)

- Illustrate the Statement.
- Prove the Statement.

**Hannah Worthington (Presentation #4): **In *neutral geometry*, given

- a circle \(\mathscr{C}_r(C)\)
- a chord \(\overline{AB}\) that is
*not*a*diameter* - a point \(D\) such that \(A-D-B\)

- Illustrate the Statement.
- Prove the Statement.

**Remark: **

is Josh's result.*Suggested Exercise 6.5#3*is both Josh's and Hannah's result.*Suggested Exercise 6.5#21*

**Jingmin Gao (Presentation #4): **In *neutral geometry*, given

- a circle \(\mathscr{C}_r(C)\)
- a chord \(\overline{AB}\) that is
*not*a*diameter*

**Hint: **Let \(M\) be the midpoint of chord \(\overline{AB}\). Use Josh's result. Then compare the length of segments \(\overline{AM}\) and \(\overline{AC}\).

(E-mail Mark for help if you need it.)

**Remark: ** Jingmin's result is the subject of ** Suggested Exercise 6.5#14**.

**Michael Cooney (Presentation #5): **In *neutral geometry*, given

- a circle \(\mathscr{C}_r(C)\)
- chords \(\overline{AB}\) and \(\overline{DE}\) that are
*not diameters*

- Illustrate the Statement.
- Prove the Statement.

(E-mail Mark for help if you need it.)

**Taylor Miller (Presentation #5): **In *neutral geometry*, given

- a circle \(\mathscr{C}_r(C)\)
- chords \(\overline{AB}\) and \(\overline{DE}\) that are
*not diameters*

- Illustrate the Statement.
- Prove the Statement.

(E-mail Mark for help if you need it.)

**Remark: ** Michael's and Taylor's results are the subject of ** Suggested Exercise 6.5#15**.

**Aoife Zuercher and Howard Bartels (Presentation #4): **Given a circle \(\mathscr{C}_r(C)\) in *neutral geometry*, it is easy to produce examples of

- a line \(L\) that
*does not intersect*the circle - a line \(L\) that intersects the circle
*exactly once*(a*tangent line*) - a line \(L\) that intersects the circle
*exactly twice*(a*secant line*)

Exercise 6.5#5 (one of your *Suggested Exercises*) is about proving that in *neutral geometry*, a line *cannot* intersect a circle *more than two times*.

The goal for your presentations is to solve this exercise. There are two cases. Each of you will solve one case. Here are the details:

Suppose that in a *neutral geometry*, a line \(L\) intersects a circle \(\mathscr{C}_r(C)\) at points \(A\) and \(B\). Let \(D\) be any other point on line \(L\). Our goal will be to show that \(D\) *does not* lie on circle \(\mathscr{C}_r(C)\).

There are two possibilities for where point \(D\) lies on line \(L\): Either (i) point \(D\) is *between* \(A\) and \(B\), or (ii) point \(D\) is *not between* \(A\) and \(B\).

**Aoife (Presentation #4): **Show that in **Case (i)**, \(CD < r\). (This tells us that \(D\) *does not* lie on circle \(\mathscr{C}_r(C)\).)

(E-mail Mark for help if you need it.)

**Howard (Presentation #4): **Show that in **Case (ii)**, \(CD > r\). (This tells us that \(D\) *does not* lie on circle \(\mathscr{C}_r(C)\).)

(E-mail Mark for help if you need it.)

The goal of ** Suggested Exercise 6.5#16** is to prove the following

**Fact: **In *neutral geometry*, given

- a circle \(\mathscr{C}_r(C)\)
- a line \(L\) that intersects the circle at \(Q\)

**Mandie Dicicco (Presentation #5): **Write the *contrapositive* of the **Fact** just stated.

(E-mail Mark for help if you need it.)

**Mark will prove the Fact.**

**Fri Apr 22 Meeting Topics**

** Statement S: ** For all segments \(\overline{AB}\) and points \(P\) in a

**Joe Durk (Presentation #5):**Is*Statement S*true? Explain why or why not.**Rachel Han (Presentation #5):**Write the*contrapositive*of*Statement S*. Is the*contrapositive*true? Explain why or why not.**Kyle Hill (Presentation #5):**Write the*converse*of*Statement S*. Is the*converse*true? Explain why or why not.**Jack Lazenby (Presentation #5):**Write the*negation*of*Statement S*. Is the*negation*true? Explain why or why not.

** Statement U: ** For all angles \(\angle ABC\) and \(\angle DBE\) in a

**Taylor Miller (Presentation #5):**Is*Statement U*true? Explain why or why not.**Jack Muslovski (Presentation #5):**Write the*contrapositive*of*Statement U*. Is the*contrapositive*true? Explain why or why not.**Logan Prater (Presentation #5):**Write the*converse*of*Statement U*. Is the*converse*true? Explain why or why not.**Sammi Rinicella (Presentation #5):**Write the*negation*of*Statement U*. Is the*negation*true? Explain why or why not.

** Statement V: ** For all trangles \(\Delta ABC\) and points \(P\) such that \(B-P-C\) in a

**Jake Schneider (Presentation #5):**Is*Statement V*true? Explain why or why not.**Helen Sitko (Presentation #5):**Write the*contrapositive*of*Statement V*. Is the*contrapositive*true? Explain why or why not.**Danielle Stevens (Presentation #5):**Write the*converse*of*Statement V*. Is the*converse*true? Explain why or why not.**Josh Stookey (Presentation #5):**Write the*negation*of*Statement V*. Is the*negation*true? Explain why or why not.

** Statement W: ** For all trangles \(\Delta ABC\) in a

**Hannah Worthington (Presentation #5):**Is*Statement W*true? Explain why or why not.**Aoife Zuercher (Presentation #5):**Write the*contrapositive*of*Statement W*. Is the*contrapositive*true? Explain why or why not.**Howard Bartels (Presentation #5):**Write the*converse*of*Statement W*. Is the*converse*true? Explain why or why not.**Jingmin Gao (Presentation #5):**Write the*negation*of*Statement W*. Is the*negation*true? Explain why or why not.

**James Foley (Presentation #5): **Consider the following statement.

** Statement Y: ** For all angles \(\Delta ABC\) and points \(P\) in a

- Is
*Statement Y*true? Explain why or why not. - Write the
*negation*of*Statement Y*. Is the*negation*true? Explain why or why not.

**Final Exam (Covering the entire course) **on Fri Apr 29, 1:00pm - 3:00pm in Gordy 311

- Your
**New List of Axioms, Definitions, and Theorems for the Final Exam** - A Calculator that has Inverse Trig Functions

- Given Quantified Conditional Statement
- Is it True? Explain
- Write Contrapositive. Is it True? Explain
- Write Converse. Is it True? Explain
- Write Negation. Is it True? Explain

- Prove injective/surjective properties of compositions of functions (Ch 1 concepts)
- Prove that some relation is an equivalence relation (Ch 1, 3, 5, 6 concepts)

- Problem about collinearity of three points (Ch 2,3 concepts)
- Given Three Points \(A,B,C\) in the Upper Half Plane
- Describe poincaré lines \(\overleftrightarrow{BA}\),\(\overleftrightarrow{BC}\)
- Draw Poincaré angle \(\angle ABC\)
- Find the measure of Poincaré Angle \(\angle ABC\). (Give an exact answer in symbols and a decimal approximation rounded to 3 decimal places.)

- Justify & illustrate given proof of theorem involving plane separation concepts (Ch 4 concepts)

- Prove Existence of an object with certain properties (Ch 3,4,5 concepts)
- Proof that certain objects are congruent (Ch 6 concepts)
- Proof that certain objects are big & small (Ch 6 concepts)

(link)

page maintained by Mark Barsamian, last updated Wed Apr 28, 2022