Help on whether a geometry solution is valid.

Help on whether a geometry solution is valid.

Let $ABC$ and $AB'C'$ be similar right angled triangles with right angles
at $C$ and $C'$, respectively. Let $l$ be the line between $C$ and $C'$,
and let $D$ and $D'$ be the points on $l$ such that $BD$ and $B'D'$ are
perpendicular to $l$.
Prove that $CD=C'D'$.
You can solve this in a pretty easy way just through a quick construction
and similar triangle ratios. But it made me ask myself if you could argue
the problem like this.
Supposing $C,A,C'$ are col-linear, then there exists a homothety centered
at $A$ that maps one triangle to the other. And $CD=C'D'=0$ for any ratio
between the two triangles. So can you say that upon rotation this
relationship will always be the same thus finishing the problem.
(Obviously I'd have to quote the properties of something or other to do
this in a proof - if it works at all).
Thanks for any help.