Three lines and three rotations using reflections.

Question

The questions are from Naive Lie Theory by John Stillwell, problem 1.5.2.-1.5.3.:

My main question is about 1.5.3., but I believe I should explain how I understood 1.5.2. For example, if we want to rotate about $P$ by $\theta$ in the diagram below, we first reflect by line $M$, then line $L$ will be flipped to the other side of line $M$. Then we reflect through this flipped line $L'$, then we get the rotation about $P$.

But I could not understand how 1.5.3. works. If I rotate about $P$ and then about $Q$, the point $R$ is moved from the original point (either by using reflections or just simply rotating them). It is because once $Q$ is moved by rotation about $P$ so that the length from the moved $Q'$ and the original $R$ becomes longer than the original distance between $Q$ and $R$, no matter how we rotate about $R$, $Q$ cannot be superimposed at $Q'$. Therefore, I could not see how the composition of the two reflections become a rotation about $R$.

I believe there is some mistake in my understanding but I could not figure it out myself. It will be helful if I could actually see each step of the proposed composition of rotations.

Answer 1

Write the two given rotations in terms of reflections at lines present in the image. Then observe that some cancellation takes place.

Answer 2

OK, I just wanted to inform why I couldn't understand. I understood the points $Q$ and $R$ as moving along with all these reflections and rotations, hence why I suggested 1.5.3. should work first with the $M$ and the flipped line $L'$. It actually meant w.r.t. the fixed points $P,Q,R$. In this case, $L$ should move first in 1.5.3., and indeed the order shows cancellation happens nicely in 1.5.4.