# 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.

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## 9 Answers

Oh boy. That, like most math, made my brain hurt. I am so not a math person. What made me love geometry was proofs. I loved the neat way everything fit.

The trouble with teaching it this way is that you’re kind of going backwards. You conclude, from the fact that a parallelogram has radial symmetry, that opposite sides and angles are congruent. However, where is your proof that a parallelogram has radial symmetry? We know this intuitively but that doesn’t cut it in a mathematical proof. You must prove the symmetry, and the tools you will use to do so include such facts as opposite sides and angles being congruent. This is why I say you’re going about it a bit backwards.

Once you prove symmetry, though, as you said you have new tools at your disposal. Knowing the complete definition and properties of symmetry, you can now draw new conclusions about a parallelogram.

Mathematics can be defined as the study of patterns. So symmetry fits right in to math – various kinds of symmetries are described by various areas of mathematics (geometry, group theory, etc.).

Even physicists can argue from symmetry: Assume the universe is isotropic (same in all directions) and homogeneous (same in all locations) and those symmetries permit surprisingly deep inferences.

When I was a physics undergraduate taking classical electrodynamics, the professor demonstrated that Maxwell’s equations, together with symmetry principles, imply special relativity and the invariance of the speed of light (though not its actual value)! Of course after Einstein, hindsight is 20–20…

There’s a theorem that among closed curves in the plane, a circle encloses maximum area for given perimeter. The proof is based on symmetry together with very basic Euclidean geometry.

[addendum, too late to append to above]

There’s a famous proposition in Euclid known *pons asinorum* (“bridge of asses”): The angles opposite the equal sides of an isosceles triangle are equal. The proof involves symmetry whereby the triangle is, in effect, lifted and turned over to demonstrate congruency.

@gasman , Are you familiar with Noether’s Theorem, that says that all the conservation laws of physics correspond to symmetries in the laws of nature? Unfortunately, I can’t follow the proof, but the result is rather mind blowing.

@LostInParadise Yes, it’s remarkable. Somehow I only learned of it a few years, despite being a physics major and studying the Lagrangian formulation of mechanics. For me that’s decades in the past. Noether stands out because there are so few women in the history of mathematics.

@ratboy, Thanks for the link. The proof for the irrationality of square root of two really is made simple and intuitive. The proof of Morley’s Theorem was a bit more difficult, but I could follow it. The one on knots lost me.

@ratboy That’s a cool geometric proof I haven’t seen before, and John Conway is one of my mathematician heroes as well. Among other things, he invented the “Game of Life,” an early cellular automaton played on primitive computers.

There’s a well known algebraic proof of the irrationality of sqrt 2, also a *reductio ad absurdum*, based on consideration of even and odd integers. It was known to the society of Pythagoreans who, according to legend, murdered its discoverer by drowning, so heretical was the nature of this theorem to these ancient mathematical mystics.

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