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Weird Area Identity

Source: 1977 USAMO Problem 2

April 4, 2010
geometryvector

Problem Statement

ABC ABC and A′B′C′ A'B'C' are two triangles in the same plane such that the lines AA′,BB′,CC′ AA',BB',CC' are mutually parallel. Let [ABC] [ABC] denotes the area of triangle ABC ABC with an appropriate ± \pm sign, etc.; prove that 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].
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