The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.
The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let be a plane intersecting a right circular cone with vertex in the curve . Call the spheres tangent to the cone and the plane and , and the circles on which the spheres are tangent to the cone and . Pick a line along the cone which intersects at , at , and at . Call the points on the plane where the sphere are tangent and . Because intersecting tangents have the same length,
(1)
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(2)
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Therefore,
(3)
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which is a constant independent of , so is an ellipse with .