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 s_{1},s_{2}...s_{n} are real numbers with
s_{1}+s_{2}+...+s_{n} = 1, then
for any vectors p_{1},...,p_{n},
t(Σs_{i}p_{i}) = Σs_{i}t(p_{i}).
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.
proofs of the lemmas
proof of the fixed point theorem
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).
proof of lemma 3
We will also need the fact that an affine transformation which fixes three
noncollinear points is the identity, this is a consequence of the uniqueness
clause in the Fundamental Theorem of Affine Geometry. It follows that,
if a nonidentity 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^{1}oros}, where r is reflection in a line, and sεA(2).
proof of strong theorem
