Three Tangents to a Sphere

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Two lines, and , are tangent to a sphere at points and , respectively. Find the line that is tangent to the sphere and intersects the other two tangents.


Call the point of tangency and let the intersection of with and be and . Determine the set of all points .


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



If and are unit vectors parallel to and , a point lies on a plane that contains the line and forms equal angles with and . There are two such planes; one is parallel to , the other to . So all are in the intersections of these two planes with the sphere, that is, on two circles on the sphere. In this Demonstration we show one of them [1, p. 224].


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

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