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Problem G5 - IMO Shortlist 2007

Source: ISL 2007, G5, AIMO 2008, TST 3, P2

July 13, 2008
geometrycircumcircleIMO Shortlist

Problem Statement

Let ABC ABC be a fixed triangle, and let A1 A_1, B1 B_1, C1 C_1 be the midpoints of sides BC BC, CA CA, AB AB, respectively. Let P P be a variable point on the circumcircle. Let lines PA1 PA_1, PB1 PB_1, PC1 PC_1 meet the circumcircle again at A′ A', B′ B', C′ C', respectively. Assume that the points A A, B B, C C, A′ A', B′ B', C′ C' are distinct, and lines AA′ AA', BB′ BB', CC′ CC' form a triangle. Prove that the area of this triangle does not depend on P P. Author: Christopher Bradley, United Kingdom
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