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:
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:
Best Answer
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