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T,R,K are collinear

Source: ItaMO 2009, p5

February 23, 2012
geometrycircumcircleprojective geometrygeometry proposed

Problem Statement

Let ABCABC be an acute-angled scalene triangle and Γ\Gamma be its circumcircle. KK is the foot of the internal bisector of ∠BAC\angle BAC on BCBC. Let MM be the midpoint of the arc BCBC containing AA. MKMK intersect Γ\Gamma again at A′A'. TT is the intersection of the tangents at AA and A′A'. RR is the intersection of the perpendicular to AKAK at AA and perpendicular to A′KA'K at A′A'. Show that T,RT, R and KK are collinear.
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