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prove that BPC<120

Source: APMO 2008 problem 1

March 22, 2008
geometryEulercircumcircleincentergeometric transformationreflectiontrigonometry

Problem Statement

Let ABC ABC be a triangle with ∠A<60∘ \angle A < 60^\circ. Let X X and Y Y be the points on the sides AB AB and AC AC, respectively, such that CA \plus{} AX \equal{} CB \plus{} BX and BA \plus{} AY \equal{} BC \plus{} CY . Let P P be the point in the plane such that the lines PX PX and PY PY are perpendicular to AB AB and AC AC, respectively. Prove that ∠BPC<120∘ \angle BPC < 120^\circ.
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