# Secant Secant Theorem

## Contents

## 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

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.