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Isogonal Conjugates: 2011 USAMO #5

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April 28, 2011
AMCUSA(J)MOUSAMOgeometrytrigonometryisogonal conjugates

Problem Statement

Let PP be a given point inside quadrilateral ABCDABCD. Points Q1Q_1 and Q2Q_2 are located within ABCDABCD such that \angle Q_1BC=\angle ABP, \angle Q_1CB=\angle DCP, \angle Q_2AD=\angle BAP, \angle Q_2DA=\angle CDP. Prove that Q1Q2‾∥AB‾\overline{Q_1Q_2}\parallel\overline{AB} if and only if Q1Q2‾∥CD‾\overline{Q_1Q_2}\parallel\overline{CD}.
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