Theorems of Ceva and Menelaus, an Illustrated Generalization
A theorem by M. Klamkin and A. Liu not only combines two famous theorem - those of Ceva and Menelaus - in a single statement but also extends both in a significant way.
Let A1A2A3 be a triangle with points B1, B2, B3 and C1, C2, C3 in sidelines A2A3, A1A3, and A1A2, respectively. Assume b1, b2, b3 and c1, c2, c3 are real numbers such that
|(*)||b1b2b3 + c1c2c3 + b1c1 + b2c2 + b3c3 = 1.|
On load, the applet displays one of the configurations known yet to the Dutch artist M. C. Escher. When the sides of ΔA1A2A3 are divided into 3, 4, and 5 segments, there are twelve such concurrences whose existence is assured by the theorem.
|What if applet does not run?|
- M. S. Klamkin, A. Liu, Simultaneous Generalization of the Theorems of Ceva and Menelaus, Mathematics Magazine, Vol 65, No 1 (February 1992), pp. 48-52