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 According to Geometer's Sketchpad... (spoiler) by Charlie)

In both the original problem and the variation, if points D, E and F are allowed to be on their respective lines (all or some) (as is necessary in the Triangles and Rectangle puzzle), the result is different. For example, in the original problem, if all the points are on the portions of the lines outside the bounds of the segments making the triangle and all the drawn lines are full infinite lines, the new triangle is 3 times the area of the original triangle.

Edited on August 23, 2005, 6:43 pm

Posted by Charlie on 2005-08-23 18:42:00