Tangent-Secant Theorem (Proof)

Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/45954

Proof:
Join points A to C and points B to C.
\[\angle BTC = \angle CTA \;{\rm{ (common)}}\]
\[\angle BCT = \angle CAT \;{\rm{ (alt}}{\rm{. \;segment \;theorem)}}\]
\[\Delta BCT \;{\rm{ is \;similar \;to }}\;\Delta CAT.\;{\rm{ (AAA)}}\]
Therefore,
$$\begin{align}
\frac{{CT}}{{AT}} & = \frac{{BT}}{{CT}} \\
\\
AT \times BT & = C{T^2} \\
\end{align}$$