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Sum of length of chords equal another

Source: CGMO 2016 Q2

August 13, 2016
geometrycircumcircle

Problem Statement

In △ABC,BC=a,CA=b,AB=c,\triangle ABC, BC=a, CA=b, AB=c, and Γ\Gamma is its circumcircle. (1)(1) Determine a necessary and sufficient condition on a,ba,b and cc if there exists a unique point P(P≠B,P≠C)P(P\neq B, P\neq C) on the arc BCBC of Γ\Gamma not passing through point AA such that PA=PB+PCPA=PB+PC. (2)(2) Let PP be the unique point stated in (1)(1). If APAP bisects BCBC, prove that ∠BAC<60∘\angle BAC<60^{\circ}.
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