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?

I think we'd all agree that triangle STU has 1/4 the area of triangle PQR; that is, it has area X/4.  Also, the smallest triangle has an area 1/4 of that, or X/16.

So the hole has area X/16; but what is the non-hole part of the donut?  Is it all of PQR except for the hole, with triangle STU being a mere aid in construction?  Then the area of the donut is 15 X / 16.  But if the outer boundary of the donut is triangle STU, then the area of the donut is 3 X / 16.

Posted by Charlie on 2006-01-01 12:53:42