I am looking for a previous reference to the following very simple geometric lemma, which I use in my paper [[arXiv:1503.03462]](http://arxiv.org/abs/1503.03462):


Let $P$ be the parabola $y=x^2$. Let $a,b,c,d$ be four points on $P$ sorted from left to right, and let $z$ be the point of intersection of the segments $ac$ and $bd$. Define the horizontal distances $p=b\_x-a\_x$, $q=d\_x-c\_x$, $r=z\_x-b\_x$, $s=c\_x-z\_x$. Then $p/q=r/s$.


This is easily proven algebraically (though I wonder whether a nice geometric proof exists...)


EDITED TO ADD: This lemma can be stated in a "power of a point"-like way: Let $P$ be the parabola $y=x^2$, and let $a$ be a point not on $P$. Let $\ell$ be a line through $a$ intersecting the parabola at points $b$ and $c$. Then the product $|a\_x-b\_x|\cdot |a\_x-c\_x|$ is independent of the choice of $\ell$.



Reference to parabola lemma
Not only power of point, but many things about circles may be defined for parabolas with vertical aces. For example, we may define the angle between lines $y=ax+b$ and $y=cx+d$ as $a-c$. Then parabola passing two points $A$, $B$ is a locus of points $P$ for which angle between lines $PA,PB$ is fixed. More advanced example: parabola passing midpoints of the sides of triangle does touch inscribed parabola of this triangle.


This analogy is sometimes called Galilean geometry. There is a book by Yaglom about it, I guess that correct English reference is


A simple non-Euclidean geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity. (Springer 1979)


