Two Solutions

Exercise. If are concurrent cevian lines for triangle , show that

(Note. The following solutions require a knowledge of Ceva' and Menelaus' Theorems as well as knowledge of cross ratios, harmonic division and complete quadrilaterals.)

Solution 1. In the figure below extend to meet in . Let the cevian lines be concurrent on a point .

Since and are menelaus points, the point is such that , or Thus, Now , and the result follows.

Solution 2. In the above figure, consider the quadrilateral . If we make this a complete quadrilateral, then and are vertices and and are diagonal points. We know that on the diagonal line there is a harmonic range consisting of the two diagonal points and the two vertices lying on the diagonal line. Hence . But and it follows that .

Bill Richardson 2010-11-12