Intersecting Chord Theorem

Theorem
The chord theorem states that if two chords, $$CD$$ and $$EF$$, intersect at $$A$$, then

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

Proof
Join C with F and E with D



then we have

$$ $$

by AA similarity we have $$\triangle FCA \sim \triangle DEA$$

$$\dfrac{CA}{FC}= \dfrac{EA}{DA}$$

$$ CA \cdot DA = EA \cdot FA$$

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