Circumcircles of Two Midpoints and an Altitude

In the triangle let and be the midpoints of the sides and and let be the foot of the altitude from to . Prove that the circumcircles of the triangles , , and have a common point and that the line passes through the midpoint of the segment .

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This Demonstration comes from problem 6 of the shortlisted problems for the 1970 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. 69.
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