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MMO 333 Moscow MO 1956 altitudes concurrent in circumcenter

Source:

August 19, 2019
geometryaltitudeconcurrentCircumcentercircumcircle

Problem Statement

Let OO be the center of the circle circumscribed around △ABC\vartriangle ABC, let A1,B1,C1A_1, B_1, C_1 be symmetric to OO through respective sides of △ABC\vartriangle ABC. Prove that all altitudes of △A1B1C1\vartriangle A_1B_1C_1 pass through OO, and all altitudes of △ABC\vartriangle ABC pass through the center of the circle circumscribed around △A1B1C1\vartriangle A_1B_1C_1.
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