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.
Theorem
Let A_{1}A_{2}A_{3} be a triangle with points B_{1}, B_{2}, B_{3} and C_{1}, C_{2}, C_{3} in sidelines A_{2}A_{3}, A_{1}A_{3}, and A_{1}A_{2}, respectively. Assume b_{1}, b_{2}, b_{3} and c_{1}, c_{2}, c_{3} are real numbers such that
(*) | b_{1}b_{2}b_{3} + c_{1}c_{2}c_{3} + b_{1}c_{1} + b_{2}c_{2} + b_{3}c_{3} = 1. |
A proof of the theorem appears in a separate page. Below is a Java applet that illustrates the theorem. You may find it useful in verifying your understanding of the barycentric coordinates.
On load, the applet displays one of the configurations known yet to the Dutch artist M. C. Escher. When the sides of ΔA_{1}A_{2}A_{3} 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? |
References
- 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
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