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BPC is a right angle for angle bisectors of ABC and ACB

Source: APMO 2011

May 18, 2011
geometrygeometric transformationreflectioncircumcirclecomplex numbersgeometry proposed

Problem Statement

Let ABCABC be an acute triangle with ∠BAC=30∘\angle BAC=30^{\circ}. The internal and external angle bisectors of ∠ABC\angle ABC meet the line ACAC at B1B_1 and B2B_2, respectively, and the internal and external angle bisectors of ∠ACB\angle ACB meet the line ABAB at C1C_1 and C2C_2, respectively. Suppose that the circles with diameters B1B2B_1B_2 and C1C2C_1C_2 meet inside the triangle ABCABC at point PP. Prove that ∠BPC=90∘\angle BPC=90^{\circ} .
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