Successful students will master a variety of concepts in Euclidean geometry. In particular:

• Students will prove theorems about lines, circles, triangles, and other geometric shapes, and learn material substantially beyond what is taught in high school.
• Students will learn some of the axiomatic approach, which builds geometry from absolute scratch and does not rely on ``intuition''. To this end, students will read selections from the most well-known classic in all of mathematics: Euclid's Elements.
• Students will learn the constructive approach. Students will learn to construct geometric figures using ruler and straightedge and to explain their constructions to others.

• Thoroughly understand what definitions and theorems are. The student will be able to precisely state definitions and theorems and understand how they are applied.
• Practice writing proofs. It is expected that the student will have some, but not a lot, of experience writing proofs. The student will gain more practice.
• Practice drawing good pictures. Diagrams are a very important part of the presentation of quantitative information, and geometry gives us an excellent opportunity to practice. It is expected that your pictures will be not only correct, but also clear and as simple as possible.

Warning. You should expect 5-8 hours of homework a week in this class, which is more than most other instructors assign; in my experience there is no other way to learn the material. Your consistent effort will certainly lead to improved understanding, and it will almost certainly lead to you earning high grades.

• Isaacs, Geometry for College Students (2001), buy it here.

• Stillwell, The Four Pillars of Geometry (2004, recommended), buy it here.

• Euclid, Elements (c. 300 BCE), available for free here.

• Hilbert, The Foundations of Geometry (1899), available for free here.

Warning. I assign a lot of homework.

The homework is intended to take 5-8 hours a week. That is a lot. Please count on making a consistent effort to do well in this class! Starting the night before is a bad idea.

If homework takes you more than 10 hours on any given week, then that is more than I intended; please let me know.

There will be at least one bonus problem on each homework, each worth one or two points, up to a maximum score of 11/10 on each week's homework. This is the only way to earn extra credit; please note that bonus problems will not be accepted late.

Graduate credit: If you want graduate credit, you are required to do at least one bonus problem from every other homework (i.e., do one from HW1 or HW2, etc.) Bonus problems on top of that count for extra credit.

Please note. You will be graded both on correctness and on quality of exposition. Indeed, a major focus of Math 531 is the ability to communicate mathematical ideas clearly. The standard is that someone who doesn't know the answer should be able to easily follow your work. In particular, please write in complete English sentences and draw clear diagrams where appropriate. Any work that is confusing, ambiguous, or poorly explained will not receive full credit.

Grading scale: A = 90+, B+ = 85+, B = 78+, C+ = 73+, C = 66+, D = 52+.

Please note that my grading scale is more generous than the usual 10-point scale. However, I am a (slightly) stricter grader than most. This is intended to balance out.

Grade component	% of grade
Two in-class exams	20% x 2
Final exam:	35%
Homework:	25%

If you have a legitimate conflict with any of the exams it is your responsibility to inform me at least a week before the exam. Otherwise makeup exams will be given only in case of documented emergency.

Late homework will be accepted once per student, up to a week late; after that, no late homework please except in case of emergency.

Academic honesty and attendance are expected of all students.

Calculators will not be needed or allowed. You will eventually require a compass and straightedge, including on exams.

• The Building Blocks of Geometry (4 weeks): We will investigate basic properties of lines, triangles, circles, angles, and other basic geometric shapes. This will likely review what you already know, but at a more advanced level and with proofs. We will cover all of Ch. 1 of Isaacs and borrow from Stillwell as appropriate.

• The Foundations of Geometry (4 weeks): Having seen the modern approach, we will then investigate the axiomatic foundations of the subject. This will take us all the way to the beginning, to Euclid. We will also read Hilbert's 1899 Foundations of Geometry, and see what Euclid left out.

• The Constructive Approach (2 weeks): This was Euclid's approach: what good are geometric shapes if we cannot build them? We will bust out the toys and construct geometric objects using compass and straightedge.

• Advanced Topics in Geometry (5 weeks): Finally, we will return to Isaacs' book and cover as much as we have time for. We will prove fascinating theorems about triangles, circles, and lines that you likely haven't seen before. Probably we will concentrate on Sections 2 and 3.

(Note: Daily topics are listed only for some days, and when in the future are subject to change.)

Homework 0, optional, due Wednesday, August 29.

If you turn this homework in, I will give feedback but the grading will be very lenient. This homework asks you to write proofs, an essential skill to review for this course.

Homework 1, due Monday, September 3. Isaacs, Chapter 1B, 2, 3, 5, 7, 8; bonus: 9. Solutions (due to Matt Boylan).

If you also hand in Homework 0 then only three problems are required from Homework 1 (your choice).

Homework 2, due Monday, September 17 (by 5:00): Isaacs, Chapter 1D, 4, 6, 7, 8, 9, 10; 1E, 1, 3. Bonus: 1D, 12, 13.

Solutions by Matt Boylan: Part 1, Part 2.

Homework 3, due Wednesday, September 26. Isaacs, Chapter 1F, 1, 2, 6, 7, 10; 1G, 2; 1H, 3, 5, 6, 8. Bonus: 1F, 12, 14; 1H, 11.

Solutions by Matt Boylan: Part 1, Part 2, Part 3.

Practice Exam 1 (solutions to be posted later).

Also, Matt Boylan's exam review sheet from 2011 (Part 1, Part 2) might be helpful. (On Part 2, only through Chapter 1G.)

Exam 1 (with solutions).

Homework 4, due Friday, October 12

Homework 5, due Wednesday, October 17. Solutions.

Homework 6, due Monday, October 29.

Exam 2, Wednesday, October 31. Study guide, practice exam with solutions.

Homework 7, due Monday, November 19: Isaacs: Ch. 2A, 1, 2, 3, 4 or 5 (bonus: do both); 2B, 1; 2C, 2 or 3 (bonus: do both); 2E, 1, 2, 3, 4 (bonus: 5).

See Matt Boylan's website for solutions.

Homework 7 bonus (one point each): For each of the Axioms of Order appearing in Group II of Hilbert, imagine that Hilbert had left this axiom out. Then, either (1) prove that this axiom follows from the other axioms, or (2) construct a ``model of geometry'' (i.e., draw a bunch of points and say which are between others) in which the other axioms are true and the omitted axiom is false.

Homework 8 (now complete), due Friday, December 7.

Solutions, except for a few problems which Matt Boylan wrote solutions to, which you can find on Matt Boylan's website.

Final exam information:

Here is a final study guide. The final will be roughly half computations and constructions, and half proofs. Here is a partial practice final, for the proofs section only. Here is the diagram. I wrote two versions of this document, and selected this one at random. You will be required to choose six out of the other seven proofs and do them, and there will be one more proof. The remainder the exam will be computations (no proofs required) and constructions.

The world's hardest easy geometry problem.