What's the angle of rotation of a product of two reflections? Let $F_1$ and $F_2$ be two arbitrary reflections about two lines in $\mathbb R^2$. 
I've been trying to work out the angle of rotation of $R_1R_2$. To this end I drew pictures in which I reflect one point first along $R_1$ then along $R_2$. Then my plan was to calculate the angle of rotation between the point and its image but the problem I ran into was that I don't have the center of rotation. 

Could someone help me and explain to me how to find the angle of $F_1
 F_2$ (the product)?

Edit If possible using a geometric argument. 


The figure gives a simple geometric answer. 
Let $O$ the fixed point of the two reflections that is the fixed point of the rotation. Than:
$P'$ is the relfection of $P$ in the line $OM$ and $P''$ the reflection of $P'$ in $ON$ and we have 
$$
\angle POM=\angle P'OM
$$
$$
\angle P'ON=\angle P''ON
$$
and $\angle P'ON+\angle P'OM= \angle MON$, so $\angle POP''=2 \angle MON$
