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?

Since S,T,U are the mid points of ¥ÄPQR so ST = ¨öQR and          UT = ¨öPQ and SU = ¨öPR and all sides ST, TU, US are parellel to RQ, QP and PR respectively.

Therefore area of ¥ÄSTU = ¨ù area of ¥ÄPQR

Let A be the mid-point of ST,B be the mid-point of TU,  and C be the mid-point of US.

SO similiarly as above

area of ¥ÄABC = ¨ù area of ¥ÄSTU

Therefore area of ¥ÄABC = (1/16)*X

and area of triangle donut  = (1/4)*X-(1/16)*X = (3/16)*X 

Posted by akash on 2006-01-01 11:16:26