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## 'Converse' printed from http://nrich.maths.org/

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?