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Sharygin CR 2019 P9

Source:

March 6, 2019
geometry

Problem Statement

Let AMA_M be the midpoint of side BCBC of an acute-angled ΔABC\Delta ABC, and AHA_H be the foot of the altitude to this side. Points BM,BH,CM,CHB_M, B_H, C_M, C_H are defined similarly. Prove that one of the ratios AMAH:AHA,BMBH:BHB,CMCH:CHCA_MA_H : A_HA, B_MB_H : B_HB, C_MC_H : C_HC is equal to the sum of two remaining ratios
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