affine congruence

An important part of the study of any geometry is the identification of the
congruence classes. We shall consider triangles, quadrilaterals and conics.

triangles in affine geometry

In euclidean geometry, we define the triangle ABC as the subset of R2
consisting of the vertices A, B and C and the sides consisting of the
points on the segments AB, BC and CA. To avoid degenerate cases,
we insist that A, B and C are non-collinear.

In affine geometry, we also have the concept of a line segment. Hence,
we can use the same formal definition of triangle, and conclude that

affine transformations map triangles to triangles

From the affine group page, we have

The Fundamental Theorem of Affine Geometry
If L =(A,B,C) and L' = (A',B',C') are lists of non-collinear points of R2,
then there is a unique element of A(2) mapping L to L'.

This has two interesting corollaries in the present context:

Corollary 1
All triangles are affine congruent.

Suppose that ABC and PQR are triangles, so (A,B,C) and (P,Q,R) are lists
of non-collinear points. By the Fundamental Theorem there is an affine
transformation t mapping A, B, C to P, Q, R, respectively. Then t maps
triangle ABC to triangle PQR, so the triangles are A(2)-congruent.

Note. The triangle ABC consists of the points of the segments AB, BC, CA.
The set of points may equally be described as the triangle ACB, BAC, BCA,
CAB, or CBA! In euclidean geometry, even if the triangles ABC and PQR are
E(2)-congruent, we can usually map A,B,C to P,Q,R only in one particular
order. Here it does not matter!

In the klein view section, we meet the idea of symmetries of a figure
in any geometry. For the case of triangles in affine geometry, we have,
as a consequence of the uniqueness clause in the fundamental theorem,

Corollary 2
For any triangle T, the A(2)-symmetry group of T is isomorphic to S3.

proof of corollary 2

Note that the situation here is quite different to that in euclidean and
similarity geometry. In those geometries, only an equilateral triangle
has six symmetries. This is so since we must be able to map any side
to any other, so the ratio of lengths must be 1 in all cases. We can also
observe that, in euclidean and similarity geometry, an isosceles triangle
has two symmetries - the identity and reflection in the bisector of the
apex angle. All other triangles have only the identity symmetry.

Once we have looked at the points fixed by a finite subgroup of A(2),
we will have a nice affine proof of the Medians Theorem.

This is rather more group theory
than geometry - it could easily
be omitted at first reading.

main affine page