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:
 * $CA \cdot CD = CB \cdot CE$

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 known as
This result is also known as the intersecting secant theorem or just the secant theorem.

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.