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prove that the line PQ bisects AD

Source: Sharygin Finals 2023 10.5

August 2, 2023
geometrySharygin Geometry OlympiadSharygin 2023

Problem Statement

The incircle of a triangle ABCABC touches BCBC at point DD. Let MM be the midpoint of arc BAC^\widehat{BAC} of the circumcircle, and PP, QQ be the projections of MM to the external bisectors of angles BB and CC respectively. Prove that the line PQPQ bisects ADAD.