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?
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 nonhole 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 20060101 12:53:42 