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An intersection of perpendicular bisectors and a weird angle condition

Source: Poland 73-3-5

March 31, 2022
geometryperpendicular bisector

Problem Statement

Let ABCABC be a triangle satisfying AB<ACAB<AC. Let MM be the midpoint of BCBC. A point PP lies on the segment ABAB with AP>PBAP>PB. A point QQ lies on the segment ACAC with ∠MPA=∠AQM\angle MPA = \angle AQM. The perpendicular bisectors of BCBC and PQPQ intersect at SS. Prove that ∠BAC+∠QSP=∠QMP\angle BAC + \angle QSP = \angle QMP.
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