Secant Secant Theorem

Theorem

Let $C$ be a point external to a circle $ABED$.

Let $CA$ and $CB$ be straight lines which cut the circle at $D$ and $E$ respectively.

Then:

$CA \cdot CD = CB \cdot CE$

Also known as the intersecting secant theorem or just the secant theorem.

Proof

Draw $CF$ tangent to the circle.

From the Tangent Secant Theorem we have that:

$CF^2 = CA \cdot CD$

$CF^2 = CB \cdot CE$

from which the result is obvious and immediate.

$\blacksquare$

Also see

This result is a generalization of the Intersecting Chord Theorem where the point of intersection of the two lines is outside the circle.

The Power of a Point Theorem is a generalization of both.