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 .

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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).
Reference
[1] D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011 p. 73.
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