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Collinear

Source: IberoAmerican Mathematical Olympiad 2004, Problem 5

September 23, 2004
geometrycircumcircleperpendicular bisector

Problem Statement

Given a scalene triangle ABC ABC. Let A′ A', B′ B', C′ C' be the points where the internal bisectors of the angles CAB CAB, ABC ABC, BCA BCA meet the sides BC BC, CA CA, AB AB, respectively. Let the line BC BC meet the perpendicular bisector of AA′ AA' at A′′ A''. Let the line CA CA meet the perpendicular bisector of BB′ BB' at B′ B'. Let the line AB AB meet the perpendicular bisector of CC′ CC' at C′′ C''. Prove that A′′ A'', B′′ B'' and C′′ C'' are collinear.
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