The ice cream cone proof is the Dandelin spheres proof that an ellipse obtained by intersecting a plane and a cone has the focal property that the sum of the distances from each point on the ellipse to two fixed foci is a constant.
The two Dandelin spheres are tangent to the plane at the foci 
and 
and to the cone along two circles.
For a point 
on the ellipse, let a generator
of the cone through 
meet these circles at 
and 
. The two tangent segments from 
to each sphere have equal lengths,
so 
and 
.
It follows that
 |
which is independent of 
. The name refers to the visual resemblance of the spheres
inside the cone to scoops of ice cream.
