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Three points on sidelines must be feet of altitudes!

Source: Sharygin Correspondence 2016

December 11, 2016
geometryisogonal conjugates3D geometry

Problem Statement

Let MA,MB,MCM_A, M_B, M_C be the midpoints of the sides BC,CA,ABBC, CA, AB respectively of a non-isosceles triangle ABCABC. Points HA,HB,HCH_A, H_B, H_C lie on the corresponding sides, different from MA,MB,MCM_A, M_B, M_C such that MAHB=MAHC,M_AH_B=M_AH_C, MBHA=MBHC,M_BH_A=M_BH_C, and MCHA=MCHBM_CH_A=M_CH_B. Prove that HA,HB,HCH_A, H_B, H_C are the feet of the corresponding altitudes.
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