Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then
a 2 + b 2 + c 2 = ab + bc + ca.
Is the converse true, and if so can you prove it? That is if a 2 + b 2 + c 2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?