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Intersections of Circumcircles and Hypotenuse

Source: 2013 CMO #3

March 31, 2013
geometrycircumcircle

Problem Statement

Let GG be the centroid of a right-angled triangle ABCABC with ∠BCA=90∘\angle BCA = 90^\circ. Let PP be the point on ray AGAG such that ∠CPA=∠CAB\angle CPA = \angle CAB, and let QQ be the point on ray BGBG such that ∠CQB=∠ABC\angle CQB = \angle ABC. Prove that the circumcircles of triangles AQGAQG and BPGBPG meet at a point on side ABAB.
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