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

 * SecantSecantTheorem.png

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.

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.