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MS =MF wanted, orthocenter, angle bisector, perpendicular, perp. bisector

Source: 2022 Austrian Federal Competition For Advanced Students, Part 2 p2

October 5, 2022
geometryequal segmentsorthocenter

Problem Statement

Let ABC ABC be an acute-angled, non-isosceles triangle with orthocenter HH, MM midpoint of side ABAB and ww bisector of angle ∠ACB\angle ACB. Let SS be the point of intersection of the perpendicular bisector of side ABAB with ww and FF the foot of the perpendicular from HH on ww. Prove that the segments MSMS and MFMF are equal.
(Karl Czakler)
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