Question

if cosA/bc + cosB/ac + cosC/ab =a^2+b^2+c^2/3k then find k
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Answer 1

In any triangle, we have

a/SinA = b/SinB = c/SinC (this is the Sin rule applicable for any triangle).

And here CosA/a = CosB/b = CosC/c

So CosA/SinA = CosB/SinB = CosC/SinC

or CotA = CotB = CotC

In a triangle this is possible only when A = B = C = 60 degrees.

So the trianle is equilateral.

And hence a = b = c = 2

Or area = [root(3)/4] * (2^2) = root(3). { Area of an equilateral triangle is [root(3)/4]*(a^2) }

Hope that helps.