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. 

Triangles are similar if and only if their angles match up.
For the triangles to be similar then, the right angle of the one triangle must match up with the right angle of the other (why?).
This means that side $d$ pairs with side $a$.
But there are still two possibilities: $b$ and $c$ can pair up with $e$ and $f$, respectively, or they can pair up with $f$ and $e$, respectively.
Your conjecture assumes $e = xb$ and $f = xc$, which is the case when $b, c$ pair up with $e, f$; but what about the other case when $b,c$ pair up with $f,e$?
