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Isogonal conjugate on a fixed circle

Source: Turkey Olympic Revenge 2023 P6

March 6, 2023
geometry

Problem Statement

In triangle ABCABC, DD is a variable point on line BCBC. Points E,FE,F are on segments AC,ABAC, AB respectively such that BF=BDBF=BD and CD=CECD=CE. Circles (AEF)(AEF) and (ABC)(ABC) meet again at SS. Lines EFEF and BCBC meet at PP and circles (PDS)(PDS) and (AEF)(AEF) meet again at QQ. Prove that, as DD varies, isogonal conjugate of QQ with respect to triangle ABC ABC lies on a fixed circle.
Proposed by Serdar Bozdag
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