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Given angles of triangle, prove third order segment equality

Source: 2018 Latvia BW TST P11

March 26, 2022
geometryangle bisector

Problem Statement

Let ABCABC be a triangle with angles ∠A=80∘,∠B=70∘,∠C=30∘\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ. Let PP be a point on the bisector of ∠BAC\angle BAC satisfying ∠BPC=130∘\angle BPC =130^\circ. Let PX,PY,PZPX, PY, PZ be the perpendiculars drawn from PP to the sides BC,AC,ABBC, AC, AB, respectively. Prove that the following equation with segment lengths is satisfied AY3+BZ3+CX3=AZ3+BX3+CY3.AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.
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