Dmytro Taranovsky
October 8, 2001

## Invertible Affine Transformations in Geometry

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

1. Check if the problem conditions are invariant under the transformation. The transformation can be used only if they are.
2. 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:

1. A congruence transformation;
2. Multiplying the y coordinate of every point by the same non-zero number: a*y-->y; or
3. 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.