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

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 known as

The Secant Secant Theorem is also known as the Intersecting Secant Theorem or just the Secant Theorem.

Also see

• Tangent Secant Theorem

• Intersecting Chords Theorem, of which this result is a generalization, where the point of intersection of the two lines is outside the circle.

• Power of a Point Theorem: a generalization of both.