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Show that the triangle is equilateral

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October 5, 2010
geometry unsolvedgeometry

Problem Statement

Show that if the sides a,b,ca, b, c of a triangle satisfy the equation 2(ab2+bc2+ca2)=a2b+b2c+c2a+3abc,2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc, then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.
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