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x^3- (a^2 + b^2 + c^2)x -2abc = 0 for quadrilateral inscribed in semicircle

Source: 2010 Saudi Arabia Pre-TST 2.4

December 28, 2021
semicircleinscribedgeometry

Problem Statement

Let AMNBAMNB be a quadrilateral inscribed in a semicircle of diameter AB=xAB = x. Denote AM=aAM = a, MN=bMN = b, NB=cNB = c. Prove that x3−(a2+b2+c2)x−2abc=0x^3- (a^2 + b^2 + c^2)x -2abc = 0.
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