Consider the symmetry group of the plane. Suppose $f$ is an element and has finite order. Show that $f$ has a fixed point. Classify all such $f$.
I'm not sure how to proceed. I can't figure out how to relate finite order with fixed points. I want someone to help me out in this.
Note that in the original question, there were two more parts to this question namely
a. Find all elements of order $2$ in this group.
b. Let $C$ be a polygon in $\mathbb R^2$ and let $f$ be a symmetry of $\mathbb R^2$. If $D=f(C)$, show that $f$ maps the centroid of $C$ to the centroid of $D$.
both of which I was able to solve. The answer to the first one is all reflections and rotation by $\pi$. These two parts may have some use in the original question, but I don't have any idea how to do that.
Hints
For (a) What does $f$ do to the midpoint of $(x, f(x))$? Then look for the set of fixed points.
For (b) the centroid is not equidistant from the vertices (look at a triangle that's not equilateral).