CAPS 2014 Problem 1

Prove that if the positive real numbers $a, b, c$ satisfy the equation $$a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),$$ then there is a triangle $ABC$ with internal angles $\alpha, \beta, \gamma$ such that $$\sin \alpha = a, \sin \beta = b, \sin \gamma= c.$$