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equilateral wanted, FP = EH . cyclic quadrilateral, altitudes , orthocenter

Source: OIFMAT I 2010 day 1 p2 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 25, 2021
Equilateralgeometrycyclic quadrilateral

Problem Statement

In an acute angle △ABC \vartriangle ABC , let AD,BE,CF AD, BE, CF be their altitudes (with D,E,F D, E, F lying on BC,CA,AB BC, CA, AB , respectively). Let's call O,H O, H the circumcenter and orthocenter of △ABC \vartriangle ABC , respectively. Let P=CF∩AO P = CF \cap AO . Suppose the following two conditions are true: ∙\bullet FP=EH FP = EH ∙\bullet There is a circle that passes through points A,O,H,C A, O, H, C Prove that the △ABC \vartriangle ABC is equilateral.
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