# Dandelin's Theorem

## Theorem

Let $\CC$ be a double napped right circular cone with apex $O$.

Let $\PP$ be a plane which intersects $\CC$ such that:

$\PP$ does not pass through $O$
$\PP$ is not perpendicular to the axis of $\CC$.

Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.

Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.

### Foci

$\SS$ and $\SS'$ are tangent to $\PP$ at the foci of $\EE$.

### Directrices

Let $\KK$ and $\KK'$ be the planes in which the ring-contacts of $\CC$ with $\SS$ and $\SS'$ are embedded respectively.

The intersections of $\KK$ and $\KK'$ with $\PP$ form the directrices of $\EE$.

## Also known as

Dandelin's theorem is also seen referred to as Dandelin's construction.

## Source of Name

This entry was named for Germinal Pierre Dandelin.