Math 465: Foundations of Geometry

Math 584: Euclidean and Non-Euclidean Geometry

Information Page, Fall 2014

A short biography of Euclid and Euclid

Here is a short note about Euclid and his "Elements".

Here are numerous proofs of the Pythagorean Theorem.

Here you can read about power of points with respect to a circle and its application to challenging high-school problems.

Here is a "proof" that all triangles are equilateral.

Nine point circle

Here are some properties of the nine-point circle.

Here and here you will find a discussion of the orthic traingle and its properties..

Here is a proof of the Simson's line theorem, and here is a discussion of the Simson line.

Here is a nice visualization of Miquel point. Here is a proof of Miguel point theorem (Pivot theorem) and also several properties of the Simson line.

Here is a discussion of Miquel point and other results due to Miquel.

Here is a discussion of Menealus's Theorem, with several proofs and additional topics.

Here is a nice article about finite geometries.

Here is a short discussion of affine and projective planes.

Here is an overview of Hilbert's axioms.

Here is a discussion of the Wallace-Gerwien-Bolyai Theorem. Here is a discussion of an analogous problem in three dimensions (often called Hilbert's third problem).

Here is a nice discussion of inversion. See various links at the bottom which all show various applications of inversions. In particular, read the one about Apollonian circles theorem, which is related to problem 39.10.

Here is another place where many interesting results related to inversion are discussed.

Here is a short discussion of inversion with many references.

Here is a very nice applet showing the Poincare model of the Hyperbolic plane. Play with it!

Here is another nice applet.

Here is a more substantial discussion and visualization of hyperbolic geometry.

Outline for review

Assignments

Assigments must be written carefully. You are allowed (even encouraged) to discuss the assigments with other students but you have to write the solutions on your own. Identical solutions will not be graded. If your solutions are not written clearly, they will not be graded and you will receive no credit.