Similar triangles question If I have a right triangle with sides $a$.$b$, and $c$ with $a$ being the hypotenuse and another right triangle with sides $d$, $e$, and $f$ with $d$ being the hypotenuse and $d$ has a length $x$ times that of $a$ with $x \in \mathbb R$, is it necessary for $e$ and $f$ to have a length $x$ times that of $b$ and $c$ respectively?

EDIT: The corresponding non-right angles of both triangles are the same. 

EDIT: This answer is now incorrect since the OP changed his question.
This is a good way of visualizing the failure of your claim:

Imagine the point C moving along the circle from A to the north pole. This gives you a continuum of non-similar right triangles with a given hypotenuse (in this case the diameter).
