Saturday, June 27, 2009

Notes on the nine-point circle, the Simson line and related topics

Mihai Caragiu - NOTES ON THE NINE POINT CIRCLE, THE SIMSON LINE AND RELATED TOPICS (Mathematics Bonus Files - April 17, 2009). ABSTRACT: These notes are based on the proofs of the nine point circle and the Simson line theorems prepared by the author for the 2006 and 2007 Summer Honors Institute camps held at the Ohio Northern University, and on the author’s 2007 October 26 talk presented at the Ohio MAA Fall Meeting held at Wittenberg University.

Tuesday, June 16, 2009

Monday, June 8, 2009

Giovanni Domenico Cassini (1625–1712)




















Giovanni Domenico Cassini
was born on June 8, 1625. He investigated the curves that are now known as the Cassini ovals defined by equations of the form




where q1 and q2 are fixed points, "dist" is the Euclidean distance in the plane, and b is a constant. The above picture displays a few Cassini ovals, with foci q1(-1, 0) and q2(1, 0), annotated with the value of b2.

In 1680, while director of the Paris Observatory he discovered the identity




satisfied by the Fibonacci numbers. This is now known as the Cassini identity. A quick proof by using determinants:





A bijective proof was provided by M. Werman and D. Zeilberger - "A bijective proof of Cassini's Fibonacci identity", Discrete Mathematics 58 (1), 109 (1986).

Also see the Cassini biography from MacTutor.