Ratios Involving Six Radii
Given a point on the side of the triangle , let and be the radii of the inscribed circles of the triangles and , and let and be the radii of the circumcircles of the triangles and . Let and denote the radii of the inscribed circle and the circumcircle of the triangle , respectively. Prove that .
The label for each radius is placed at the center of the circle of which it is the radius.
This Demonstration comes from problem 8 of the shortlisted problems for the 1970 International Mathematical Olympiad (IMO).
 D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011 p. 69.