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midpoint lies on perpendicular bisectors, circumcenters related

Source: Czech-Polish-Slovak Junior Match 2013, Individual p5 CPSJ

March 14, 2020
geometryperpendicular bisectormidpointCircumcenter

Problem Statement

Point MM is the midpoint of the side ABAB of an acute triangle ABCABC. Point PP lies on the segment ABAB, and points S1S_1 and S2S_2 are the centers of the circumcircles of APCAPC and BPCBPC, respectively. Show that the midpoint of segment S1S2S_1S_2 lies on the perpendicular bisector of segment CMCM.
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