I am trying to prove that the composition of two reflections in non-parallel lines (i.e. lines that intersect) is a rotation.
From observation I can see that using $L_1$ as the $x$-axis and $L_2$ as the $y$-axis (so the angle $\theta = \frac{\pi}{2}$) that the composition $r_{L_1L_2}$ is a rotation around the origin (the point of intersect) by $\pi$ ($= 2\theta$).
How can I prove that the composition of any reflections (not just my example) is a rotation? Any pointers are appreciated.
Note $L_i^t=-L_i$. Moreover, $L_i^2=I$, since $L_i$ is a reflection. Therefore $L_1L_2(L_1L_2)^t=L_1L_2[(-L_2)(-L_1)]=L_1L_2L_2L_1=L_1IL_1=L_1^2=I$, and similary $(L_1L_2)^tL_2L_1=I$. Thus $(L_1L_2)^{-1}=(L_1L_2)^t$ and $\mathrm{det}(L_1L_2)=1$. Hence $L_1L_2$ is a rotation.
Note that in the case of the lines to be parallel, then the rotation is simply the identity matrix (trivial rotation).