I have two lines $m$ and $l$ in the affine space $A^{2}$ and a point $M$ not on $l$ or $m$. Prove that there exist two unique points $A$ on $l$ and $B$ on $m$ so that $M$ is the middle of $A$ and $B$. I'm trying to prove this analytical and synthetic.
I started with the analytical way.
So let's say $l$ and $m$ intersect in a point $p_{0}$. And then... Normally you can give coördinates to points that are given but now the only thing given ( and not to prove) are the two lines. I tried to give $A=(a_{1},a_{2})$ and $B=(b_{1},b_{2})$ and then I had the line $l=p_{0}A$ and the line $m=p_{0}B$. When I gave $p_{0}=(0,0)$ and $M=(1,1)$ I got that $a_{1}=b_{1}$ and $a_{2}=b_{2}$. This is really wrong but I don't see my fault.
The synthetic way I tried with finding a homothetie $H$ with centrer $p_{0}$ so that $H(A)=B$. But again I'm stuck and don't think this is the right way to do it. Someone who can help me :)