Intersecting chords theorem

Dependencies: Vertical angles are equal, Inscribed angles on same arc are equal; angle in a semicircle is right (inscribed angles on the same arc are equal), AA similarity.

Statement

Suppose two chords $AB$, $CD$ in $\odot O$ meet at an interior point $P$. Then the products of the four chord segments satisfy

In other words, as long as $P$ is fixed inside the circle, for any chord through $P$, the product of the two segments into which $P$ cuts the chord is the same constant.