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


Also see

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