Nathaniel Emerson, USC

Title: The Pythagorean Theorem and Similar Triangles: Historical and Modern Examples

Abstract: The Pythagorean Theorem and similar triangles are two fundamental mathematical results. They have been known for at least 
3800 years. Nonetheless, mathematicians continue to find new ways combining these principles.

We with discuss two Ancient Greek proofs of the Pythagorean Theorem. The first an elementary proof of the Pythagorean Theorem using similar triangles. The second the famous proof from Euclid’s Elements. Proofs like these had a major influence on the development of Medieval Algebra.   

High-school students A.~Jackson and C.~Johnson recently published several surprising proofs of the Pythagorean Theorem. We will discuss one of their proofs that uses infinitely many similar triangles. This proof uses geometry to obtain Calculus-like results.