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Angle bissector

Source: French TST 2004 pb.2; German contest (Bundeswettbewerb) 2003, 1st round

May 25, 2004
trigonometrygeometryangle bisectorgeometry proposed

Problem Statement

Let ABCDABCD be a parallelogram. Let MM be a point on the side ABAB and NN be a point on the side BCBC such that the segments AMAM and CNCN have equal lengths and are non-zero. The lines ANAN and CMCM meet at QQ. Prove that the line DQDQ is the bisector of the angle ∡ADC\measuredangle ADC. Alternative formulation. Let ABCDABCD be a parallelogram. Let MM and NN be points on the sides ABAB and BCBC, respectively, such that AM=CN≠0AM=CN\neq 0. The lines ANAN and CMCM intersect at a point QQ. Prove that the point QQ lies on the bisector of the angle ∡ADC\measuredangle ADC.
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