Let ABCDE be a convex pentagon such that the
triangles ABC, ACD, and ADE are similar.
The diagonals BD and CE intersect in point P.

Prove that the ray AP bisects side CD.

1. Construct the circumcircles of ADE and ACE. Note that these intersect at P.
2. Construct the circumcircle of PDC. This intersects the ray AP at K.
3. Construct the triangle CDK. It can trivially be established that this is congruent to ACD.
4. It is now easily seen that CD is a diagonal, and AQ a half-diagonal, of parallellogram ACKD, from which the required result follows at once.

Posted by broll on 2010-10-19 06:47:11