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Geometric Inequality

Source: Iranian National Olympiad (3rd Round) 2006

September 21, 2006
inequalitiestrigonometrygeometryincentergeometric transformationreflectioncircumcircle

Problem Statement

In triangle ABCABC, if L,M,NL,M,N are midpoints of AB,AC,BCAB,AC,BC. And HH is orthogonal center of triangle ABCABC, then prove that LH2+MH2+NH2≤14(AB2+AC2+BC2)LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})
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