some fixed point theorems

 We have seen that the affine symmetry group of a triangle always has exactly six elements. One way to prove the Medians Theorem is to show that all six fix a point, and that this point must lie on each median. Part of this is a special case of a remarkable 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. The proof actually computes a suitable P. If Q is any point, then we may take P with position vector p = (1/n)Σg(q), where the sum is over the elements of G, n is the order of G, and q is the position vector of Q. The fixed point need not be unique. If r denotes reflection in a line L, then each element of the group {e,r} fixes every point of L. Here, if Q is any point, then it is easy to see that P is the foot of the perpendicular from Q to L. Lemma 1 If tεA(2) and s1,s2...sn are real numbers with s1+s2+...+sn = 1, then for any vectors p1,...,pn, t(Σsipi) = Σsit(pi). This Lemma is useful also in proving a result about transformations which fix more than one point Lemma 2 If tεA(2) fixes points P and Q, then it fixes every point of the line PQ. Of course, the theorem applies to finite subgroups of E(2), a subgroup of A(2). In this case, we can be more precise since we know that an element of E(2) which fixes two points of a line L is either the identity or reflection in L. This is proved as in Lemma 1 for euclidean geometry. The Euclidean Fixed Point Theorem If G ≠ {e} is a finite subgroup of E(2),then either (1) there is a unique point fixed by every element of G, or (2) G = {e,r}, where r is reflection in a line. If we have more than one fixed point, we have a line fixed by every element of G. But then, by the preceeding remarks, G contains only the identity and reflection in this line. In affine geometry, there are other subgroups fixing a line, but we shall see that these are still of order two. A useful step towards this is Lemma 3 If tεA(2) fixes every point of the line L, and sεA(2), then u = sotos-1 fixes every point of s(L). We will also need the fact that an affine transformation which fixes three non-collinear points is the identity, this is a consequence of the uniqueness clause in the Fundamental Theorem of Affine Geometry. It follows that, if a non-identity element of A(2) fixes the points of a line, then it cannot fix any other point. The Strong Affine Fixed Point Theorem If G ≠ {e} is a finite subgroup of A(2),then either (1) there is a unique point fixed by every element of G, or (2) G = {e,s-1oros}, where r is reflection in a line, and sεA(2).