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AK+ BL+CM =0 in vectors => ABC equilateral

Source: 2004 Estonia National Olympiad Final Round grade 12 p3

March 25, 2020
vectorEquilateralgeometry

Problem Statement

Let K,L,MK, L, M be the feet of the altitudes drawn from the vertices A,B,CA, B, C of triangle ABCABC, respectively. Prove that AK→+BL→+CM→=O→\overrightarrow{AK} + \overrightarrow{BL} + \overrightarrow{CM} = \overrightarrow{O} if and only if ABCABC is equilateral.
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