MAT 515 Geometry for teachers - Fall 2016

Syllabus,  schedule and homework.

Slides, geogebra files, and other materials can be found here.

This schedule will be regularly updated. It is your responsibility to check it accordingly.

Homework is due every Wednesday. Underlined problems in should be handed in.

When Topics  # Homework, Exams, Remarks.
8-29 General Information.
Here are the first pages of the book.
Discussion about axiom systems.
History of geometry
1B Congruent Triangles
1C Angles and parallel lines
1D Parallelograms
0 Homework 0: Fill this form and play Euclid the game,  and/or , Euclidea (you can also download it in a smart phone or tablet).
9-5 1E  Area

1 Sept 5th, No class - Labor Day
1B - 1,3, 4
1D - 1, 3, 4, 5,12, 13,14
9-12 1F Circles and arcs
1H Similarity
2 In this link you'll find corrections to known errors in the book.
1E: 1, 2, 3, 4  (In IE.3 add hypothesis the triangle is isosceles)
Read and understand the proof of Problem 1.14      
9-19 2A The circumcircle
2B The centroid
2C The Euler line, Orthocenter, and Nine-Point circle
3 HW3:
1F: 2, 3, 5, 6, 7, 10, 11, 12,14. In the problem 1.F11, you need to produce a graph of the problem in Geogebra and share it with  me (you will  need to create a Geogebra account for that) . Think of this graph as a prop you would use to explain a student the problem.
From Wednesday Sept 21st (including Sept 21st) the class will be held in 4-130 Math Tower.
1H:  2, 3, 5, 8, 10.
9-26 2C The Euler line, Orthocenter, and Nine-Point circle (cont.)
2D Computations
2E The incircle

4 By popular demand! Here is "our" song: That's mathematics, and here is his author, Tom Lehrer, signing, and here are the lyrics.
1H:  5, 10.
2A: 1, 5
2B: 1, 2, 4
2C: 1, 2.

5 2C: 4, 6.
2D: 1, 2
2E: 1, 3, 4, 5
-In Geogebra, create a triangle and label all its "special" points  (centroid, feet of altitudes, midpoints of sides, etc. They are 12:  9 in the nine point circle and three in the Euler line). Verify the Nine Point Circle Theorem and the Euler Line Theorem. Mark and label the center of the Nine Point Circle. Verify the relations between the (pairwise) distances between orthocenter, circumcenter and centroid.  Share the file with me in Geogebra.

First Midterm: Chapters 1 (except Section 1G) and Chapter 2 (Except Sections 2F, 2G and 2H)

Here is a nice page about different geometries, including the bear problem. (The whole website is quite interesting)
10-10 G Morley's theorem
2F Exscribed circles
2H Optimization in triangles
6 -In Geogebra, create  a triangle, and find its incircles and the three excircles (Figure 2.21 of the textbook). Of course,  the only "free" objects of the graph must be the three vertices. Verify Theorem 2.34 of the textbook by computing the sum of the inverses of the radius of the excircles minus the inverse of the radius of the incircle.
-In Geogebra, create a triangle and construct the three angle trisectors. Highlight Morley's triangle and verify it is equilateral.

Share both files with me.

Also, the problem 5 for the midterm : Prove that the nine-point circle bisects any segment connecting the orthocenter to a point on the circumcircle. (Hint: Recall that the radius of the nine-point circle is half the radius of the circumcircle)

Spherical geometry.
7 2F: 1, 2.
2G: 1.
2H: 1.

Submit the three underlined problems above and problems A and B below.
A. Determine what is the figure formed by the intersections of the four pairs of adjacent angle quadrisectors of the angles of a square (As in Morley theorem, consider each intersection of two angle quadrisectors which are closer to a side). Prove your findings. 
B . Read the proof of Morley's theorem in the textboox and write up a version of that proof in your own words.
Extra credit:  Find a formula for the length of the sides of the equilateral triangle determined by the angle trisectors of a triangle. Your formula should be in terms of the angles of the triangle and the circumradius.

Also, the problem 5 for the midterm (if you did not do it yet): Prove that the nine-point circle bisects any segment connecting the orthocenter to a point on the circumcircle. (Hint: Recall that the radius of the nine-point circle is half the radius of the circumcircle)

Morley Theorem
10-24 Isometries.
Here and here you'll find good introductions to isometries and here are notes by Oleg Viro, revised by Olga Plamevskaya.

Isometries case 1
Isometries case 2

In class, we will work with this handout.
8 This homework contains a bit of review of previous topics.
1. In class we discussed under which hypothesis  the SSA criteria for congruence of triangles holds. In this problem, you are asked to list all possible cases as we did in class,  and prove why in each of these cases the SSA criteria hold or does not hold.
2. Use the spherical law of cosines to calculate the  great-circle distance NY (40.7oN,74oW) to Tashkent (41.3oN, 69.2oE).
3. Prove that if two sides of a triangle are unequal, then the larger side has a larger angle opposite to it.
4. . Prove that if two angles of a triangle are unequal, then the larger angle has a larger side opposite to it.
5. In  triangle ABC, A line AD is drawn, where D is a point in BC.  From B and C two perpendiculars to line AD are dropped, intersecting AD in E and F. If M is the midpoint of BC, prove that FME is isosceles.
6. In any quadrilateral, the lines joining the midpoints of each pair of opposite sides and the line joining the midpoints of the diagonals are concurrent.
10-31 Isometries.

9 Problems for this week are here. Hand in problems 2, 5,11, 12, 13, 14 (In problem 14, you only need to produce an appropriate list of symmetries, no need of proofs in this problem);
11-7 Similarities. Notes by notes by Oleg Viro, revised by Olga Plamevskaya
Interesting Geogebra  resources for geometry.
Stretch rotation example
Stretch reflection example

Problems for this week are here.

11-14 6A Rules of the game
6B Reconstructing triangles
11 Second midterm (focused on Sections 2F, 2G, 2H, Isometries. )
Homework for this week is here. Please hand in these problems on Monday Nov 21st.  Remember to make two sets, one containing problems 1 to 5, the other containing problems 6-7 and, if you want problem 1 from HW 10.

11-21 6C Tangents

No homework this week.
11-23 No class  Thanksgiving Break
11-28 6D Three hard problems
6E Constructible numbers
6F Changing the rules
12 6A 1, 2, 3, 4, 5, 6.
6B 1, 2, 3, 4. ( (hint for 1 and 3: when is an inscribed angle on a circle is  right?)
12-5 Extra topics and review
We will work with this handout from the book "Project Origami" by Thomas Bull

Final Exam: Thursday, Dec. 15, 8:30pm-11:00pm