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relation between sides

Source: Swiss IMO TST 2016. Problem 8

July 27, 2017
geometrybisectorperpendicular

Problem Statement

Let ABCABC be a triangle with AB≠ACAB \neq AC and let MM be the middle of BCBC. The bisector of ∠BAC\angle BAC intersects the line BCBC in QQ. Let HH be the foot of AA on BCBC. The perpendicular to AQAQ passing through AA intersects the line BCBC in SS. Show that MH×QS=AB×ACMH \times QS=AB \times AC.
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