English: Diagram for a proof of the intercept theorem.

Assume ${\displaystyle AC}$ and ${\displaystyle BD}$ are not parallel. Then the parallel line to ${\displaystyle AC}$ through ${\displaystyle D}$ intersects ${\displaystyle SA}$ in ${\displaystyle B_{0}\neq B}$. Since ${\displaystyle |SB|:|SA|=|SD|:|SC|}$ is true, we have
${\displaystyle |SB|={\frac {|SD||SA|}{|SC|}}}$
and on the other hand from claim 2 we have
${\displaystyle |SB_{0}|={\frac {|SD||SA|}{|SC|}}}$.
So ${\displaystyle B}$ and ${\displaystyle B_{0}}$ are on the same side of ${\displaystyle S}$ and have the same distance to ${\displaystyle S}$, which means ${\displaystyle B=B_{0}}$. This is a contradiction, so the assumption could not have been true, which means ${\displaystyle AC}$ and ${\displaystyle BD}$ are indeed parallel ${\displaystyle \square }$