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O_1O_2_|_CF , line connecting circumcenters, starting with isosceles

Source: 7th XMO (China) p1 https://artofproblemsolving.com/community/c3192772_geometry_regional_china

April 7, 2024
geometryperpendicular

Problem Statement

As shown in the figure, it is known that BC=ACBC = AC in ABCABC, MM is the midpoint of ABAB, points DD and EE lie on ABAB satisfying ∠DCE=∠MCB\angle DCE = \angle MCB, the circumscribed circle of △BDC\vartriangle BDC and the circumscribed circle of △AEC\vartriangle AEC intersect at point FF (different from point CC), point HH lies on ABAB such that the straight line CMCM bisects the line segment HFHF. Let the circumcenters of △HFE\vartriangle HFE and △BFM\vartriangle BFM be O1O_1 and O2O_2 respectively. Prove that O1O2⊥CFO_1O_2\perp CF. https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png
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