MATH 461 - Affine and Projective Geometry

The class meets TuTh 1:00 - 2:15pm in LGRT 141.

Professor: Jenia Tevelev

Office: LGRT 1235E. Office hours are TuW 2:30-4:00

E-mail: tevelev(at)math.umass.edu

Textbook: The four pillars of geometry by John Stillwell. We will cover pretty much the whole book except for the last section. Take lecture notes because I will often give examples and proofs that are not contained in the book.

As in any math class, it is very important to understand everything written in the textbook and covered in lectures. Read slowly and take your time understanding all definitions, examples, and proofs. Don't take anything on faith. Use as a motto a famous phrase of Alexandre Grothendieck: "I never once doubted that I would eventually succeed in getting to the bottom of things." Every time you read a statement of a theorem or lemma, pause and think about it before reading the proof. Do you know definitions of all concepts mentioned in the statement? What makes this statement non-trivial? Have you seen similar statements before? How would you prove it? When you read the proof, try to isolate its main idea. Remember that each problem is unique but there aren't that many different proof techniques. A trick used in the proof of a theorem can often be re-used in the proof of a homework problem. Reading a mathematical textbook is not always easy. Sometimes you will be under impression that two consecutive sentences are not logically connected. If this happens, the author probably wants you to make this logical connection yourself.

The homework will be posted on the class website. It will be collected each week on Thursday. Most of the problems will require writing proofs. Write them in an essay style (like proofs in the book) using complete English sentences. Check that everything you are saying is rigorously justified. You will make the grader's life easier by providing as many details as possible.

There will be two in-class midterms (October 5 and November 9) and occasional 15-minute quizzes (I will e-mail the class before every quiz).

The final grade will be 20% midterm 1, 20% midterm 2, 20% final exam, 30% HW, 10% quizzes. Two worst HW grades will be dropped. All grades will be posted on Moodle.

HW assignments:

   HW due on             Sections to read              Homework problems       Comments   
September 14 Chapter 1 + lecture notes. 1.2.3, 1.3.3, 1.3.5, 1.4.3, 1.4.4
I. Given a circle, construct the center of this circle
II. Given a circle C and points A and B, construct a point on C equidistant from A and B. Is there only one such point?
III. Construct a right isosceles triangle circumscribed around a given circle.
IV. Given a circle and a line, construct a tangent line to the circle parallel to the given line.
V. Given a circle, construct an inscribed rectangle such that the angle between its diagonals is 30 degrees.
Lecture slides.
Euclid's Elements: An interactive web-based version of the famous "Elements".
Homework solutions.
September 21 Chapter 2 + lecture notes. 2.2.2, 2.3.1, 2.3.2, 2.3.3, 2.5.2, 2.7.5
I. Let A be a point outside a circle. Draw lines AX, AY tangent to the circle at X and Y. Prove that AX=AY.
II. Let ABCD be a quadrilateral such that angles ACB and ADB are equal. Show that all vertices of this quadrilateral lie on a circle.
III. Prove that every triangle has an inscribed circle and that the center of that circle is the intersection point of all three angle bisectors of angles of the triangle.
IV. Prove that every triangle has a circumscribed circle and that the center of that circle is the intersection point of all three perpendicular bisectors of sides of the triangle.
Homework solutions.
October 3 Lecture notes Some of the 15 problems in this worksheet are simple and some could be a bit tricky. Choose and solve 6 problems from the worksheet. (So different students can solve different subsets). You are welcome to try all problems but you are not allowed to submit more than 6. All of them will be graded. Quiz on September 28.
Review of Euclidean geometry on October 3.
Midterm on October 5. Solutions for midterm.
October 12 Chapter 3 (3.1-3.5). 3.2.4, 3.2.5, 3.2.6, 3.4.1, 3.4.2, 3.4.3, 3.5.1, 3.5.2
I. A quadrilateral ABCD has vertices given by intersection points of lines y=kx+b, y=kx-b, y=mx+b and y=mx-b. Find coordinates of the intersection point of diagonals of the quadrilateral ABCD.
II. The line y=kx+k+1 (with k>0) intersects coordinate axes at points A and B. Find the smallest possible area of the triangle ABO (where O is the origin).
October 19 Chapter 3 (3.6-3.8) + lecture notes. 3.6.1, 3.6.2, 3.6.3, 3.7.1, 3.7.2, 3.7.3
I. Prove that every isometry of the plane takes circles to circles.
II. Suppose circles S1 and S2 of the same radius R intersect at points A and B. Let M and N be intersection points of S1 and S2 with the perpendicular bisector of the segment AB. Suppose, in addition, that M and N lie on the same side of the line AB. Prove: AB2+MN2=4R2.
III. Suppose a quadrilateral has an axis of symmetry. Show that either this quadrilateral is a trapezoid or the axis of symmetry is a diagonal of the quadrilateral.
IV. Construct an equilateral triangle with vertices on three given parallel lines.
October 26 Chapter 4 + lecture notes. TBA Quiz on October 24.
November 2 Chapter 5+ lecture notes. TBA
November 9 No reading assignment No HW due Review of analytic geometry on November 7.
Midterm on November 9
November 16 Chapter 6 + lecture notes. TBA
November 30 Chapter 7 + lecture notes. TBA
December 7 Lecture notes. TBA Final Exam Review on December 12