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touchpoint of incircle, orthocenter and midpoint of altitude collinear wanted

Source: 2022 Oral Moscow Geometry Olympiad grades 8-9 p6

April 17, 2022
geometrycollinearcollinearity

Problem Statement

In an acute non-isosceles triangle ABCABC, the inscribed circle touches side BCBC at point T,QT, Q is the midpoint of altitude AKAK, PP is the orthocenter of the triangle formed by the bisectors of angles BB and CC and line AKAK. Prove that the points P,QP, Q and TT lie on the same line.
(D. Prokopenko)