Secant Secant Theorem

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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.

$\blacksquare$


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.