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**by Jerzy Kocik (Southern Illinois University) and Andrzej Solecki (Universidade Federal Brazil)**

**Award:** Lester R. Ford

**Year of Award:** 2010

**Publication Information:** *The American Mathematical Monthly*, vol. 116, no. 3, March 2009, pp. 228-237.

**Summary:** The authors give a presentation of trigonometry's central theorems: in a few brief pages, decorated with repeated versions of the same simple figure consisting of an acute triangle set within its inscribed circle - each of the figures only slightly modified to highlight some particular relation among its pieces. Ptolemy's theorem, the law of sines, the nine-point circle, the standard sum and difference formulas for the trigonometric functions, these and a host of other well-known (and less well-known) results spring forth geometrically in these diagrams and in a few simple equations.

**About the Authors:** (From the Prizes and Awards booklet, MathFest 2010)

**Jerzy Kocik** studied theoretical physics in Wroclaw, Poland, but he strayed into mathematics when he realized that it explores many realities, while physics is confined to one… He is currently an associate professor at Southern Illinois University at Carbondale. His main research interests lie in Lie algebras and differential geometry. Recently, he has given in to the charms of Apollonian circle packing. Nothing thrills him more than encountering bridges between the simple and the deep, between the obvious and the concealed.

When **Andrzej Solecki** decided to study mathematics at Wroclaw University, he heard that the place was okay but why mathematics? The answer was that maths would not depend on geopolitics, it would not demand much time, it could be fun and one could play with it all alone. It took him a long time to discover that the answer was only 50% right.

Publication Date:

Wednesday, August 11, 2010