I am looking for a previous reference to the following very simple geometric lemma, which I use in my paper [arXiv: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 $\ell$.