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Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Converse

Age 14 to 16 Challenge Level:

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?