If X is the area of any trianglePQR and you make a hole in the center of this triangle by first joining the midpoints of each side to create smaller triangleSTU and then again joining the midpoints of triangleSTU 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 midpoint of ST,B be the midpoint of TU, and C be the midpoint 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 20060101 11:16:26 