If X is the area of any triangle-PQR and you make a hole in the center of this triangle by first joining the midpoints of each side to create smaller triangle-STU and then again joining the midpoints of triangle-STU to create the hole in the center of this triangle, it would create a donut something like this:

          Q 
         / \
        /   \
       /     \
    S /_______\ U
     / \  A  / \
    /   \/_\/   \
   /     \ /     \
P /_______V_______\ R
          T

What is the area of the triangle donut in terms of X?

(In reply to My interpretation by Charlie)

Taking the practical rather than the mathamatical view -

The outer boundary of the donut is the largest triangle PQR and the "hole" the smallest central triangle. Two reasons.

1. Using triangle STU as a boundary results in a hole extending to the edge of the donut - and it falls to pieces - thus no donut.

2. By "Vernons Law" - Always choose the donut with the most possible dough and the least possible hole.

Posted by Vernon Lewis on 2006-01-01 15:47:03