Dandelin spheres are two spheres tangent internally to a cone and also to a plane intersecting the cone. For an elliptical section, the two spheres lie on opposite sides of the intersecting plane.
The spheres can be used to show that the intersection of the plane with the cone is an ellipse. This argument is informally known as the ice cream cone proof. It proves that the conic section definition of an ellipse agrees with the focal definition in which the sum of the distances from a point to two fixed foci is constant.
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 spheres are tangent 
and 
. The two tangent segments from 
to each sphere have the same length,
so
 |
(1)
|
 |
(2)
|
Therefore,
 |
(3)
|
which is a constant independent of 
, so 
is an ellipse with semimajor
axis 
.
