Locus of Centers of Spheres

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The points and lie on the same side of the plane (colored gray), with not parallel to . What is the locus of the centers of the spheres through and that are tangent to ?

Contributed by: Izidor Hafner (April 2017)
Open content licensed under CC BY-NC-SA



Let the point be the intersection of and the straight line through . Let be the tangent point of the sphere with . Since , lies on the circle with center and radius . The center of the sphere is the intersection of the line through that is orthogonal to and the plane orthogonal to through the midpoints of and .

So the locus of centers of spheres lies on a cylinder orthogonal to with radius and center .


[1] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.

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