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prove that 2 points are the feet of the altitudes

Source: Oral Moscow Geometry Olympiad 2015 grades 8-9 p6

August 8, 2019
geometryangle bisectoraltitudescyclic quadrilateral

Problem Statement

In the acute-angled non-isosceles triangle ABCABC, the height AHAH is drawn. Points B1B_1 and C1C_1 are marked on the sides ACAC and ABAB, respectively, so that HAHA is the angle bisector of B1HC1B_1HC_1 and quadrangle BC1B1CBC_1B_1C is cyclic. Prove that B1B_1 and C1C_1 are feet of the altitudes of triangle ABCABC.
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