Can the perspective of symmetry make high school geometry more interesting?

I admit to being fascinated by the concept of symmetry. I like the way it relates the artistic to the mathematical. It can also provide a short intuitive way of understanding things.

After working on a problem I found on the Web, I realized that a parallelogram is symmetric with respect to a 180 degree rotation about the center of a diagonal. That is, the 180 degree rotation places the parallelogram on top of itself. From this one fact, the following immediately at an intuitive level.

1. Opposite sides of a parallelogram are congruent and parallel.
2. Opposite angles of a parallelogram are congruent.

I was thinking how much more interesting (to me, at least, and I hope to others as well) this makes the humble parallelogram. Mention is made of transformations in some high school geometry classes. Transformations are an idea from the 19th century that can liven the study of Euclidean geometry, which is otherwise taught pretty much the same as it was 2000 years ago.

I did a Web search to see if anyone else felt the way I did and found the following PowerPoint presentation I would be interested in your comments on it.

For anyone interested, here is the original problem that got me thinking about transformations. A diagonal of a parallelogram divides it into two triangles. Show that the circumcenters of the two triangles are equidistant from the diagonal.

I tried to use Euclidean geometry to solve this and gave up. Then I realized that a 180 degree rotation of the parallelogram about the center of a diagonal maps the parallelogram on top of itself. Now the proof could be done intuitively. The circumcenter of a triangle is the intersection of the perpendicular bisectors of the sides. Rotating the parallelogram 180 degrees places the perpendicular bisectors of one triangle onto those of the other. It therefore follows that their intersections are 180 degree rotations of each other. Now it is an easy proof to show that the 180 degree rotation of a point is equidistant from any line through the center of rotation.