Three Equal Circles in a Semicircle
- C. A. Pickover, A Passion for Mathematics, John Wiley & Sons, 2005, p. 82
Join the centers of three circles, as shown, and extend one line to the point of tangency of two circles.
The proof needs just one application of the Pythagorean theorem. Let r and R be the radii of the small circles and the semicircle. In the right triangle in the diagram, one of the legs equals r, the other to 2r, and the hypotenuse to R - r, giving a relation:
which translates into
Solving for R/r, we get
As the minus sign would produce a negative ratio, we settle for the sign plus: