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October 8, 2001
Transformations are very useful in geometry. They are used as follows:
The most frequently used set of transformations consist of all transformations that leave distance invariant, called congruence transformations.
Invertible affine transformation is:
The transformation leaves many conditions invariant, specifically:
The following sets of figures can be transformed into each other using the transformation:
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.