Intersecting Chord Theorem

Theorem
Let $CD$ and $EF$ both be chords of the same circle.

Let $CD$ and $EF$ intersect at $A$.

Then $CA \cdot AD = EA \cdot AF$.

Also known as the chord theorem.


 * If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

Alternative Proof
Join $C$ with $F$ and $E$ with $D$, as shown in this diagram:



Then we have:

By Triangles with Two Equal Angles are Similar we have $\triangle FCA \sim \triangle DEA$.

Thus: