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midpoint lies on circumcircle, incircle and a third circle related

Source: 2017 Singapore MO Open Round 2

March 17, 2020
geometrycircumcircleincirclemidpointcircles

Problem Statement

The incircle of △ABC\vartriangle ABC touches the sides BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. A circle through AA and BB encloses △ABC\vartriangle ABC and intersects the line DEDE at points PP and QQ. Prove that the midpoint of ABAB lies on the circumircle of △PQF\vartriangle PQF.
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