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perpendiculars and midpoints

Source: 2019 MEMO Problem T-5

August 30, 2019
geometryperpendicular bisectorcircumcirclememoMEMO 2019sinus

Problem Statement

Let ABCABC be an acute-angled triangle such that AB<ACAB<AC. Let DD be the point of intersection of the perpendicular bisector of the side BCBC with the side ACAC. Let PP be a point on the shorter arc ACAC of the circumcircle of the triangle ABCABC such that DP∥BCDP \parallel BC. Finally, let MM be the midpoint of the side ABAB. Prove that ∠APD=∠MPB\angle APD=\angle MPB.
Proposed by Dominik Burek, Poland
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