The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

Except for translations, all of the transformations above can be represented by matrix equations. These are linear transformations, because they are consistent with vector addition and scalar multiplication. This means that for any linear transformation `f` and for vectors `u` and `v`, we have `f` (`u` + `v`) = `f`(`u`) + `f` (`v`), and for any scalar `r`, `f`(`r`` u`) = `r` `f`(`u`). The geometric consequences of these rules are very useful: a linear transformation takes a line to a line, and takes midpoints to midpoints. For the first of these, suppose that `u`, `v`, and `w ` are collinear points in the plane. That means that the vector from `u` to `v` is parallel to the vector from `v` to `w`. In other words, `v` − `u` = `r` (`w` − `v`) for some scalar `r`. Now apply `f` to both sides of this equation. The properties of linear transformations allow us to distribute `f` across the vector subtractions, and to factor `r` in or out. So simplifying leads to `f`(`v`) − `f`(`u`) = `r` (`f` (`w`) − `f` (`v`)). This shows that `f`(`u`), `f`(`v`), and `f` (`w`) are also collinear.

As for midpoints, we need only observe that these are the same as averages. To see this, suppose that `m` is the point midway between `u` and `v`. That means that the vector from `u` to `m` is the same as the vector from `m` to `v`. So we derive the equation `m` − `u` = `v` − `m`, and hence 2`m` = `u` + `v`. Thus we obtain `m` = (`u` + `v`)/2, showing that the midpoint of `u` and `v` is the same as the average of `u` and `v`. Now apply `f` and we see `f`(`m`) = (`f`(`u`) + `f`(`v`))/2. This proves that `f` takes the midpoint of `u` and `v` to the midpoint of `f`(`u`) and `f`(`v`). By the same sort of logic, `f` takes a point 1/3 of the way along a line segment to an image point 1/3 of the way along the image segment, and similarly for any other fraction, as well.

When you combine a linear transformation with a translation, the result is no longer linear. These are referred to as *affine* transformations. Geometrically, it is clear that affine transformations also map lines to lines and midpoints to midpoints. To see this, separate the affine transformation into a linear transformation followed by a translation. The linear part maps lines to lines and midpoints to midpoints. The translation simply shifts an image by a fixed amount, but does not in any way distort the parts of the image. So lines are still mapped to lines, and midpoints are still mapped to midpoints.

These properties of linear and affine transformations are helpful in the context of Marden's Theorem. In particular, we use them to prove that every triangle has a unique inscribed ellipse that is tangent to the sides at the midpoints.