Three Equal Segments from the Altitudes of a Triangle
In the triangle let the feet of the altitudes be , , and . The circle with diameter meets and at two points other than to form a segment of length ; similarly define and . Prove that .
The segments of equal length are drawn in blue.
This Demonstration comes from problem 19 of the shortlisted problems for the 1971 International Mathematical Olympiad (IMO).
 D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011 p. 73.