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I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher dimensions. It generalizes easily: the radius of the circles of cone/sphere intersections in $\mathbb{R}^3$ (below, red) are equal:
   EyeBall
What I am wondering is if there is a sense in which some form of this theorem generalizes to other objects: axis-aligned cubes, ellipsoids, or other shapes. Or does the theorem in some sense characterize spheres? If anyone has seen this addressed previously, I'd appreciate a pointer. Thanks!


(Added). This seems to work for squares/cubes:
   EyeSquare

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    $\begingroup$ I think I would have enjoyed this image much more had I not read "eyeball theorem" in the title! $\endgroup$ Dec 18, 2013 at 1:02
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    $\begingroup$ And I would probably not have opened the question without the eyeball reference in the title! That is lovely, the theorem (in the plane) and the image. $\endgroup$
    – Asaf Karagila
    Dec 18, 2013 at 1:14
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    $\begingroup$ Of course this also works for axis-aligned ellipses of the same eccentricity. $\endgroup$ Dec 18, 2013 at 19:30
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    $\begingroup$ One generalization to squares fails. If you consider the lines connecting the centers of the squares to the visible corners of the other square, these don't have the same size "arcs." For example, axis-aligned squares of sides $2$ and $4$ whose centers are distance $10$ apart in the direction of an axis produce arcs of sizes $4/9$ and $1/2$. Your diagram for squares shows secant lines from the centers of squares to the centers of sides, not to the corners that I think would be analogues of tangent lines. $\endgroup$ Dec 19, 2013 at 3:22
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    $\begingroup$ Unfortunately this doesn't work if we rotate the squares $45$ degrees. $\endgroup$ Dec 22, 2013 at 17:58

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I have introduced some variants of the Eyeball theorem and also seems to admit generalizations in 3D. And as if that were not enough the Archimedean twins have been brought together with these theorems. See link below

http://geometriadominicana.blogspot.com/2014/03/praying-eyes-theorem.html

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