In the classic problem you are given a triangle ABC with points D on AB, E on BC, and F on AC such that |AD|=2|DB|, |BE|=2|EC|, and |CF|=2|FA|. The lines AE, BF, and CD enclose a triangle inside triangle ABC. You are to find the area of this enclosed triangle relative to that of ABC. The answer is 1/7.

What if everything is the same except |BE|=|EC| and |CF|=3|FA|. What is the area of the enclosed triangle relative to that of ABC?

(In reply to re: According to Geometer's Sketchpad... (spoiler) by Charlie)

If D is on the extended line AB and F on the extended line CA, rather than on their respective segments (E has to be the midpoint of segment BC; the numbers don't work out otherwise), the new triangle FED has an area 25/12 that of triangle ABC.

However, if for example, D is on segment AB but F is again on the extended line CA, not on segment CA, the new triangle, DEF, is 16/15 the area of ABC.

Posted by Charlie on 2005-08-23 20:19:18