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(PQD) bisects segment BC

Source: 2021 Saudi Arabia Training Lists p16 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 5, 2022
bisects segmentgeometry

Problem Statement

Let ABCABC be an acute, non-isosceles triangle with circumcenter OO, incenter II and (I)(I) tangent to BCBC, CACA, ABAB at D,E,FD, E, F respectively. Suppose that EFEF cuts (O)(O) at P,QP, Q. Prove that (PQD)(PQD) bisects segment BCBC.
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