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trigonometric existence of triangle given equation w sides

Source: Czech-Polish-Slovak Match 2014 day 1 P1

September 29, 2017
Trigonometric Equationspolynomial equationTrianglegeometry

Problem Statement

Prove that if the positive real numbers a,b,ca, b, c satisfy the equation a4+b4+c4+4a2b2c2=2(a2b2+a2c2+b2c2),a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2), then there is a triangle ABCABC with internal angles α,β,γ\alpha, \beta, \gamma such that sin⁡α=a,sin⁡β=b,sin⁡γ=c.\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.
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