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HMMT Feb 2023 team p9

Source:

February 20, 2023

Problem Statement

Let ABCABC be a triangle with AB<ACAB < AC. The incircle of triangle ABCABC is tangent to side BCBC at DD and intersects the perpendicular bisector of segment BCBC at distinct points XX and YY. Lines AXAX and AYAY intersect line BCBC at PP and QQ, respectively. Prove that, if DP⋅DQ=(AC−AB)2DP \cdot DQ = (AC-AB)^2 then AB+AC=3BC.AB + AC = 3BC.
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