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<PBC + <PCB = <BAC if PB/PC = PC_1/PB_1 = AB/AC

Source: 2006 VMEO III Juniors 10.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

September 11, 2021
geometryanglesratioprojections

Problem Statement

Given a triangle ABCABC (AB≠ACAB \ne AC). Let P P be a point in the plane containing triangle ABCABC satisfying the following property: If the projections of P P onto ABAB,ACAC are C1C_1,B1B_1 respectively, then
PBPC=PC1PB1=ABAC\frac{PB}{PC}=\frac{PC_1}{PB_1}=\frac{AB}{AC} or PBPC=PB1PC1=ABAC\frac{PB}{PC}=\frac{PB_1}{PC_1}=\frac{AB}{AC}.
Prove that ∠PBC+∠PCB=∠BAC\angle PBC + \angle PCB = \angle BAC.
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