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)
Okay. Then at
you are probably better off going ahead and multiplying everything out:
Now the and terms cancel so, bring everything involving m and n to the left side, everything not involving m and n to the right side,
Unfortunately, we need to get two equations out of this one. What we can do is separate x and y by taking, first to get
so that
and then take to get
so that
.
Do you see that we can do that- treat the x and y coordinates separately?
I'm afraid that I can't see that we can do that.
Clearly, it is very convenient to make , but I'm not sure why we can say this. Surely, then, , if , since this was an assumption made in its derivation. Therefore, having made a similar assumption for the y-coordinates, the desired result for the midpoint only holds if and , which would mean that we have a contradiction: the points cannot be distinct.
I realise that this argument must be wrong somewhere, as I am happy with the derivation via similar triangles, so I'd love it if you could show me the error of my ways.
(Also, just for perfection, you have a stray minus two in the numerator of your last line!)