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)
|
 |
(2)
|
Therefore,
 |
(3)
|
which is a constant independent of 
, so 
is an ellipse with 
.
