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trigonometry trivia

Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 1

November 15, 2009
trigonometrygeometrycircumcirclegeometry proposed

Problem Statement

Let P P be a point in triangle ABC ABC, and define α,β,γ \alpha,\beta,\gamma as follows: \alpha\equal{}\angle BPC\minus{}\angle BAC,   \beta\equal{}\angle CPA\minus{}\angle \angle CBA,   \gamma\equal{}\angle APB\minus{}\angle ACB. Prove that PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.
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