Prove that every translation or rotation of $\Bbb E^2$ is a composite of rotations about $P$ and $Q$. Prove that every translation or rotation of $\Bbb E^2$ is a composite of rotations about $P$ and $Q$.

Lemma 1. For $d$ with $0\le d\le 2\operatorname{dist}(P,Q)$, there exists a translation by $d$ in a suitable direction, that can be achieved.
Proof. Rotating around $P$ by $\alpha$ and around $Q$ by $-\alpha$ produces a translation where the distance (and direction) depends continuously on $\alpha$. For $\alpha=0$, the distance is $0$, and for $\alpha=\pi$ it is $2\operatorname{dist}(P,Q)$. The claim follows from the intermediate value theorem. $\square$
Lemma 2. Every translation can be achieved.
Proof. Let $d$ be the distance of the translation $\tau$. For $n$ big enough, $\frac dn$ is between $0$ and $2\operatorname{dist}(P,Q)$. By lemma 1, there exists a translation $\tau'$ by a distance $\frac dn$ that can be achieved. Then so can the $n$-fold repetition of $\tau'$, which is a translation by $d$. If the directions of $\tau'$ and $\tau$ differ by $\alpha$, one obtains $\tau$ by rotating around $P$ by $\alpha$, then applying $\tau'$ repeatedly $n$ times, then rotating around $P$ by $-\alpha$. $\square$
Lemma 3. Every rotation can be achieved.
Proof. A rotation by $\alpha$ around $X$ is obtained by first translating $X$ to $P$ (as in lemma 2), then rotating around $P$ by $\alpha$, then translating $P$ to $X$ again. $\square$.
