An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Let the radius of the circle be r. This implies that the radius
of the semicircle is 2r. The area of the semi circle is $1/2 \times
\pi \times(2r)^2$, which is twice the area of the small circle.
This problem is taken from the UKMT Mathematical Challenges.