Suppose that there are two distinct points, and , with coordinates and , respectively. At the midpoint of these points lies the point , with coordinates . Find and in terms of and .

I know that the solutions are and , but I'm not sure why.

I've tried two applications of Pythagoras' Theorem, stating that by definition , so that

.

Given the contents of a couple of the brackets, this looks promising, but I'm not sure about how to tease out the desired result from here, or even if it's possible to do so. Any ideas much appreciated. (Btw, I realise that a simpler proof using similar triangles is possible, and is documented here: http://www.mathhelpforum.com/math-he...ula-proof.html)