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given angle relationships in triangle, prove equality

Source: Croatian MO 2004 4th Grade P2

April 9, 2021
geometry

Problem Statement

Points PP and QQ inside a triangle ABCABC with sides a,b,ca,b,c and the corresponding angle α,β,γ\alpha,\beta,\gamma satisfy ∠BPC=∠CPA=∠APB=120∘\angle BPC=\angle CPA=\angle APB=120^\circ and ∠BQC=60∘+α\angle BQC=60^\circ+\alpha, ∠CQA=60∘+β\angle CQA=60^\circ+\beta, ∠AQB=60∘+γ\angle AQB=60^\circ+\gamma. Prove the equality (AP+BP+CP)3⋅AQ⋅BQ⋅CQ=(abc)2.(AP+BP+CP)^3\cdot AQ\cdot BQ\cdot CQ=(abc)^2.
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