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concurrency, <APE =<BAC, < CQF =< BCA (2013 Kyiv City MO Round2 11.4)

Source:

August 5, 2020
geometryequal anglesconcurrencyconcurrent

Problem Statement

Let H H be the intersection point of the altitudes AP AP and CQ CQ of the acute-angled triangle ABC ABC . On its median BM BM marked points E E and F F so that ∠APE=∠BAC \angle APE = \angle BAC and ∠CQF=∠BCA \angle CQF = \angle BCA , and the point E E lies inside the triangle APB APB , and the point F F lies inside the triangle CQB CQB . Prove that the lines AE AE , CF CF and BH BH intersect at one point.
(Vyacheslav Yasinsky)
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