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?

One way to prove that the area does indeed quarter on each iteration is to look at the semi-perimeter formula for the area of a triangle

A = Sqrt( s (s-a) (s-b) (s-c) ) where s = (a+b+c)/2.

Halving the lengths of each of the sides, halves each of the four terms inside the Sqrt, (1/2)^4.  Taking the square root leaves (1/2)^2 = 1/4 A.

Posted by goFish on 2006-01-02 12:03:34