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Regional Olympiad - FBH 2010 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

September 27, 2018

geometryidentity

Problem Statement

It is given acute triangle

ABC

with orthocenter at point

H

. Prove that

AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}

where

a

b

and

c

are sides of a triangle, and

h_a

h_b

and

h_c

altitudes of

ABC