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Prove an area

Source: INMO 1991 Problem 2

October 3, 2005
geometrytrigonometry

Problem Statement

Given an acute-angled triangle ABCABC, let points A′,B′,C′A' , B' , C' be located as follows: A′A' is the point where altitude from AA on BCBC meets the outwards-facing semicircle on BCBC as diameter. Points B′,C′B', C' are located similarly. Prove that A[BCA′]2+A[CAB′]2+A[ABC′]2=A[ABC]2A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2 where A[ABC]A[ABC] is the area of triangle ABCABC.
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