Most general linear transformation which leaves the origin fixed and preserves all distances? I'm working on the following problem from Ahlfors (complex analysis):

Prove that the most general (linear) transformation which leaves the origin fixed and preserves all distances is either a rotation or a rotation followed by reflexion in the real axis.

I have the part about it being a rotation. Roughly,

Let $Sz = \frac{az+b}{cz+d}$. $S(0)=0$ implies that $b=0$. $|Sz|=|z|$ implies $|cz+d|=|a|=constant$ for all $z$, so that $c=0$ and $|a|=|d|$. Together they imply $Sz = \frac{a}{d} z$ where $|a/d| = 1$.

I'm not really sure where the reflexion in the real axis part of the question comes in. Is "reflexion in the real axis" the mapping $x+iy \mapsto -x+iy$ ? Would that not mean that it is no longer a linear transformation?
My suggestion, 

S(z)=az/d. Take T(z)=-az/d and R(z)=-z. 
 Then T is a rotation, R is a reflexion in the real axis and RT(z)=S(z).
 So S is a rotation followed by reflexion in the real axis.
My suggestion, 

S(z)=az/d. Take T(z)=-az/d and R(z)=-z. 
 Then T is a rotation, R is a reflexion in the real axis and RT(z)=S(z).
 So S is a rotation followed by reflexion in the real axis.