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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 $\ell$.

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    $\begingroup$ I don't have enough rep to comment... 2 ideas : 1. It is like intersecting chords ratio, may be a good start 2. On the site www-home.math.uwo.ca/~mdawes/woma/potw_back.html prob. #980421. About colinears points. But i am not sure that is a good start. $\endgroup$
    – user70598
    Apr 16, 2015 at 14:14
  • $\begingroup$ The parabola lemma is a limit case of the intersecting chords theorem: If you take 4 points on a circle very close to the bottom part of the circle, and you affinely stretch the picture vertically, then you get almost a parabola. Since the chords were almost horizontal, their length was very close to their $x$-projection. Unfortunately, taking a limit does not count, in my opinion, as being simpler than a straigntforward algebraic calculation, so this is not a "simple" proof. $\endgroup$
    – Gabriel Nivasch
    Apr 17, 2015 at 6:01
  • $\begingroup$ BTW, this limit argument can translate any length property on a circle to an $x$-projection property on a parabola. E.g.: Let $p$ be a point below the parabola, and let the two tangents to the parabola through $p$ pass through points $a$ and $b$ on the parabola. Then $|p_x-a_x| = |p_x-b_x|$. $\endgroup$
    – Gabriel Nivasch
    Apr 17, 2015 at 6:11

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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)

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  • $\begingroup$ Thanks! Very interesting. I'll try to take a look at the book... $\endgroup$
    – Gabriel Nivasch
    Jun 4, 2015 at 17:39
  • $\begingroup$ I see that the angle-as-slope definition also makes sense in light of the limiting argument that I mentioned above: For small $\alpha$ we have $\alpha \approx \tan\alpha$ $\endgroup$
    – Gabriel Nivasch
    Jun 4, 2015 at 17:44

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