Suppose that there are two distinct points, $\displaystyle A $ and $\displaystyle B $, with coordinates $\displaystyle (x_1, y_1) $ and $\displaystyle (x_2, y_2) $, respectively. At the midpoint of these points lies the point $\displaystyle M$, with coordinates $\displaystyle (m, n)$. Find $\displaystyle m $ and $\displaystyle n $ in terms of $\displaystyle x_1, y_1, x_2 $ and $\displaystyle y_2 $.

I know that the solutions are $\displaystyle m = \frac{x_1 + x_2}{2} $ and $\displaystyle n = \frac{y_1 + y_2}{2} $, but I'm not sure why.

I've tried two applications of Pythagoras' Theorem, stating that by definition $\displaystyle AM = MB \Rightarrow AM^2 = MB^2 $, so that

$\displaystyle \begin{array}{ccc}(n - y_1)^2 + (m - x_1)^2 & = & (y_2 - n)^2 + (x_2 - m)^2 \\

(n - y_1)^2 - (y_2 - n)^2 & = & (x_2 - m)^2 - (m - x_1)^2 \\

(n - y_1 + y_2 -n)(n - y_1 - y_2 +n) &=& (x_2 -m + m - x_1)(x_2 -m -m + x_1) \\

(y_2 - y_1)(2n - y_1 - y_2) &=& (x_2 - x_1)(-2m + x_1 + x_2) \end{array} $.

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)