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Prove that the circumcircles have a common point

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September 23, 2010
geometrycircumcircleEulerpower of a pointIMO Shortlist

Problem Statement

In the triangle ABCABC let B′B' and C′C' be the midpoints of the sides ACAC and ABAB respectively and HH the foot of the altitude passing through the vertex AA. Prove that the circumcircles of the triangles AB′C′AB'C',BC′HBC'H, and B′CHB'CH have a common point II and that the line HIHI passes through the midpoint of the segment B′C′.B'C'.
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