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Question: Let XYZ be an equilateral triangle inscribed in C. If α, β, γ denote the distances of D from vertice...

Let XYZ be an equilateral triangle inscribed in C. If α, β, γ denote the distances of D from vertices X, Y, Z respectively, the value of product (β + γ - α) (γ + α - β) (α + β- γ), is

A

0

B

αβγ8\frac{αβγ}{8}

C

α3+β3+γ3−3αβγ6\frac{α^3 + β^3 + γ^3 - 3αβγ}{6}

D

None of these

Answer

0

Explanation

Solution

Let α,β,γ\alpha, \beta, \gamma be the distances of point D from the vertices X, Y, Z of an equilateral triangle inscribed in circle C. If point D lies on the circumcircle C, then by Pompeiu's theorem, one of the distances is the sum of the other two. Assume, without loss of generality, that α=β+γ\alpha = \beta + \gamma. The product is given by (β+γ−α)(γ+α−β)(α+β−γ)(\beta + \gamma - \alpha) (\gamma + \alpha - \beta) (\alpha + \beta - \gamma). Substituting α=β+γ\alpha = \beta + \gamma, the first factor becomes (β+γ−(β+γ))=0(\beta + \gamma - (\beta + \gamma)) = 0. Thus, the entire product is 0×(γ+α−β)×(α+β−γ)=00 \times (\gamma + \alpha - \beta) \times (\alpha + \beta - \gamma) = 0.