= Solution
First use the preceding <isometry> $t$ to align the first two points, and put $R_3=t(P_3)$. It remains to align $R_3$ with $Q_3$ while fixing $Q_1,Q_2$. Use <orthonormal coordinates> in which $Q_1=(0,0)$ and $Q_2=(d,0)$, $d>0$, and write $R_3=(r,s)$, $Q_3=(u,v)$. The prescribed <distances> give
$$
r^2+s^2=u^2+v^2,\qquad (r-d)^2+s^2=(u-d)^2+v^2.
$$
Subtracting yields $r=u$, and then $s^2=v^2$. Therefore $Q_3$ is either $R_3$ or its <reflection> across the line through $Q_1,Q_2$. In the first case take $s=t$; in the second compose $t$ with that <reflection>. Both constructions fix the first two images and send the third point to its required image. The argument also covers three collinear points, when the two possible third images coincide. Thus \b[all three points can be matched by one plane isometry].
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