**by Charles I. Delman, Gregory Galperin**

**Award:** Carl Allendoerfer

**Year of Award:** 2004

**Publication Information:** *Mathematics Magazine*, Vol. 76 (2003), pp. 15-32

**Summary:** Everyone knows that the sum of the angles of a triangle formed by three lines in the plane is 180 degrees, but is this still true for curvilinear triangles formed by the arcs of three circles in the plane? This paper gives a complete analysis of the situation, showing along the way the insights that can be gained by approaching the problem from several points of view and at several levels of abstraction.

**About the Authors**

**Charles Delman** is Professor of Mathematics at Eastern Illinois University. He received his Bachelor’s in mathematics from Harvard and his Ph.D. from Cornell, under the guidance of Alan Hatcher. Before coming to EIU, he taught at The Ohio State University and Pitzer College. His mathematical interests include low-dimensional topology, classical geometry, and dynamical systems.

**Gregory Galperin** is currently Professor of Mathematics at Eastern Illinois University. He received his Ph.D. from the University of Moscow under the tutelage of prominent twentieth century mathematician Andrei N. Kolmogorov. Dr. Galperin’s Ph.D. thesis concentrated on dynamical systems with local interaction. He has published more than 50 mathematical articles on billiards and other dynamical systems. Dr. Galperin has been an Alexander von Humboldt fellow since 1994. He has served as Coordinator at the 42nd International Mathematical Olympiad in Washington, D.C., and was the Deputy Leader of the USA team at the 44th International Mathematical Olympiad in Japan, 2003.

**Subject classification(s)**: Geometry and Topology