proof of the fixed point theorem

The Affine Fixed Point Theorem
If G is a finite subgroup of A(2), then there is a point P fixed by every element of G.

Let Q be any point on the plane, and let P = (1/n)Σg(Q), where the sum is over the
elements of G, and n is the order of G. This is of the form considered in Lemma 1 with
all the si equal to 1/n.
By Lemma 1, for any h in G, h(P) = Σhog(Q). But since G is a group, the set {hog}
is precisely the set G. Thus h(P) = P for all h in G.

fixed points theorems page