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All PP' lines pass through a fixed point

Source: Iran TST 2011 - Day 4 - Problem 2

May 14, 2011
vectorgeometrygeometric transformationhomothety

Problem Statement

Let ABCABC be a triangle and A′,B′,C′A',B',C' be the midpoints of BC,CA,ABBC,CA,AB respectively. Let PP and P′P' be points in plane such that PA=P′A′,PB=P′B′,PC=P′C′PA=P'A',PB=P'B',PC=P'C'. Prove that all PP′PP' pass through a fixed point.
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