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angle bisector wanted inside a regular pentagon

Source: 2020 VWO Flanders MO p3

August 8, 2020
geometryangle bisectorpentagon

Problem Statement

The point MM is the center of a regular pentagon ABCDEABCDE. The point PP is an inner point of the line segment [DM][DM]. The circumscribed circle of triangle △ABP\vartriangle ABP intersects the side [AE][AE] at point QQ (different from AA). The perpendicular from PP on CDCD intersects the side [AE][AE] at point SS. Prove that PSPS is the bisector of ∠APQ\angle APQ.
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