The usual mantra is 'three points defined a circle (or line)'. This is usually the quickest way to find the answer.
We'll use the fact that $1,3,2+i$ are in the original circle.
(a) $1\mapsto \infty$ so it is a line. $3 \mapsto \frac{1}{2}$, and $2+i \mapsto \frac{i}{1+i} = \frac{1}{2}(1+i)$. Hence we have a line through $\frac{1}{2},\frac{1}{2} + \frac{i}{2}$. This is clearly the line $\mathrm{Re} w = \frac{1}{2}$. To check which side of the line is which, note that $z=2$ is in the interior, and $2 \mapsto 0$ so the region is $\mathrm{Re} w \le \frac{1}{2}$
(b) I'll leave to you.
(c) You could do this as the above too. Alternatively, note $$|z-2| = 1 \iff (z-2)(z^\star-2) = 1 \iff 1 - 2/z - 2/z^\star+3/zz^\star = 0$$ and then using $w=1/z$ we find $$3ww^\star - 2w -2w^\star + 1 = 0 \iff \left(w-\frac 2 3 \right)\left(w^\star-\frac 2 3 \right) = \frac 1 9$$ which gives the result. Again, one point is enough to tell you where the interior is.