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IMO Shortlist 2011, G4

Source: IMO Shortlist 2011, G4

July 13, 2012
geometrycircumcirclesymmetryIMO Shortlisthomothetyradical axisgeometry solved

Problem Statement

Let ABCABC be an acute triangle with circumcircle Ω\Omega. Let B0B_0 be the midpoint of ACAC and let C0C_0 be the midpoint of ABAB. Let DD be the foot of the altitude from AA and let GG be the centroid of the triangle ABCABC. Let ω\omega be a circle through B0B_0 and C0C_0 that is tangent to the circle Ω\Omega at a point X≠AX\not= A. Prove that the points D,GD,G and XX are collinear.
Proposed by Ismail Isaev and Mikhail Isaev, Russia
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