# Intersecting Chord Theorem for Conic Sections

## Theorem

Consider a right circular cone $\CC$ with opening angle $2 \alpha$ whose apex is at $O$.

Consider a slicing plane $\PP$, not passing through $O$, at an angle $\beta$ to the axis of $\CC$.

Let the plane $OAA'$ through the axis of $\CC$ perpendicular to $\PP$ intersect $\PP$ in the line $AA'$.

Let $P$ be an arbitrary point on the intersection of $\PP$ with $\CC$.

Let $PM$ be constructed perpendicular to $AA'$.

Then:

$PM^2 = k \cdot AM \cdot MA'$

where $k$ is the constant:

$k = \dfrac {\map \sin {\beta + \alpha} \map \sin {\beta - \alpha} } {\cos \alpha}$