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citerion / converse of a property of altitudes 1991 Kyiv City MO 10.2

Source:

July 19, 2021
geometryaltitudesorthocenter

Problem Statement

In an acute-angled triangle ABCABC on the sides ABAB, BCBC, ACAC, the points C1C_1, A1A_1, and B1B_1 are marked such that the segments AA1AA_1, BB1BB_1, CC1CC_1 intersect at some point OO and the angles AA1CAA_1C, BB1ABB_1A, CC1BCC_1B are equal. Prove that AA1AA_1, BB1BB_1, and CC1CC_1 are the altitudes of the triangle.
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