Two-secants product equality

Depends on: AA similarity, Inscribed angles on same arc are equal; angle in a semicircle is right, Linear pair sums to 180°.

Statement

Let $\odot O$ be a given circle and $P$ a point outside the circle. Draw two secants through $P$: one meeting the circle at $A$ and $B$ ($A$ closer to $P$, $B$ farther from $P$), the other meeting the circle at $C$ and $D$ ($C$ closer to $P$, $D$ farther from $P$). Then the products of the two segments cut off on each secant by the circle are equal:

In other words, the "power" $PA \cdot PB$ of an exterior point with respect to the circle is independent of the chosen secant — every secant through $P$ gives the same constant.