Intersecting Chord Theorem

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

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

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

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



Then we have:

$$ $$

By AA similarity we have $$\triangle FCA \sim \triangle DEA$$.

Thus:

$$ $$