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# Converse

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### Consecutive Squares

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Challenge Level

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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?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.