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Dmytro Taranovsky

October 8, 2001

## Invertible Affine Transformations in Geometry

Transformations are very useful in geometry. They are used
as follows:

- Check if the problem conditions are invariant under the
transformation. The transformation can be used only if they
are.
- Use the transformation to change to change the problem into a
simpler one.

The most frequently used set of transformations consist of all
transformations that leave distance invariant, called congruence
transformations.

Invertible affine transformation is:

- A congruence transformation;
- Multiplying the y coordinate of every point by the same non-zero
number: a*y-->y; or
- Any combination of 1 and 2.

The transformation leaves many conditions invariant, specifically:

- Colinearity or coplanarity of a set of points: Lines
are transformed into lines; planes-into planes; parallel lines or
planes--into parallel lines or planes.
- Ratio of lengths for the segments on the same or parallel
lines.
- Ratio of areas on the same or parallel planes.
- Ratio of volumes if the space is three-dimensional.
- Center of mass is transformed into center of mass.

The following sets of figures can be transformed into each other
using the transformation:

- All tetrahedrons and thus all triangles.
- All parallelepipeds and thus all parallelograms.

The properties above can be easily proved.

**Example:** For a tetrahedron, a bimedian is a segment
from a vertex to the intersection of medians on the opposite face.
Prove that for every tetrahedron, all bimedians intersect at one
point that divides each bimedian in a 3:1 ratio starting at the
vertex; moreover, the intersection is the center of mass of the
tetrahedron, and it divides the tetrahedron into 4 tetrahedrons of
equal volume.

**Solution:** All of the conditions are invariant under
all invertible affine transformations. Since all tetrahedrons
are equivalent under the transformation, the tetrahedron can be
transformed into regular tetrahedron of side length 1 without loss of
generality. The theorem (except 3:1 ratio) follows from the
symmetry. The 3:1 ratio can be easily verified for the regular
tetrahedron.