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O_1O_2 _|_ CF

Source: 4th XMO (China) p https://artofproblemsolving.com/community/c3192772_geometry_regional_china

April 18, 2024
perpendicularperpendicularitygeometry

Problem Statement

As shown in the figure, it is known that BC=ACBC= AC in △ABC\vartriangle ABC, MM is the midpoint of ABAB, points DD, EE lie on ABAB such that ∠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, O2O_2 respectively. Prove that O1O2⊥CFO_1O_2 \perp CF. https://cdn.artofproblemsolving.com/attachments/8/c/62d4ecbc18458fb4f2bf88258d5024cddbc3b0.jpg
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