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circumcenter of A_1B_1C_1 is incenter of ABC, AA_1 = BB_1 = CC_1 = R,

Source: Tournament of Towns 2020 oral p2 (15 March 2020)

May 17, 2020
circumcirclegeometryincentercircumradiusaltitudes

Problem Statement

At heights AA0,BB0,CC0AA_0, BB_0, CC_0 of an acute-angled non-equilateral triangle ABCABC, points A1,B1,C1A_1, B_1, C_1 were marked, respectively, so that AA1=BB1=CC1=RAA_1 = BB_1 = CC_1 = R, where RR is the radius of the circumscribed circle of triangle ABCABC. Prove that the center of the circumscribed circle of the triangle A1B1C1A_1B_1C_1 coincides with the center of the inscribed circle of triangle ABCABC.
E. Bakaev
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