Intersecting Secants

In this entry, I’m going to be discussing how mathematicians tend to approach the world, and why we need better PR.

I’m currently teaching High School Geometry. Here is what the book has to say about the “Segment of Chords Theorem”: “If two chords intersect in a circle, then the products of the lengths of the chord segments are equal.” And here is the “Secant Segments Theorem”: “If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.” (Source: Glencoe Geometry, 2012 CCSS edition, pages 750 and 752.)

There are also pictures and examples. The introductory “Why?” example involves determining the radius of a round cake based on the remnant that remains after people have cut it in a way people don’t normally cut round cakes. There are some more interesting examples in the text, such using the measurement and the height of a rainbow to determine its radius.

It’s been a long time since I’ve taken High School Geometry myself, so I don’t recall if I was ever exposed to these theorems before. I do know that my first reaction, seeing these with fresh mathematician eyes, was one of excitement: This was a really cool concept. Put a point inside or outside a circle, then draw two secant lines. The product of the distances between the point and the intersection points will be the same along each secant line.

My next  impulse was to try to prove it. I thought about it for a little bit, trying to think of a way to do so. I didn’t come up with one on my own; looking it up felt like cheating, but once I decided to do so, I realized how rudimentary the proof was. I’ll summarize that below.

But at some point, I realized that this was a perfect example of the difference between how a mathematician sees the world and how we present our knowledge. Students are famous for asking, “When are we ever going to use this in life?” And fair enough, there are some vaguely useful examples in the text of how this might be used. But, truth be told, pragmatics wasn’t top on my own list of why this was interesting.

As a mathematician, I love to see the interweaving of theorems and observations and such. Some of them have practical use, some of them are just downright pretty. For me, these theorems are just downright pretty. As I’ve been writing this post, it occurs to me that I can generalize even further. Consider this statement:

Draw a circle and a point, so that you can draw a line through the point that intersects the circle in two places. The product of the distances between the point and each intersection will always be the same.

It doesn’t matter if the point is inside the circle, on the circle, or outside the circle. It’s always true. You can fire up GeoGebra, drop in a few objects, and rotate a secant line to your heart’s content: It will always be true.

This is how my inner mathematician feels about that.

And now, let’s go back to what the students are presented. Two different theorems, separated by several pages (actually, three theorems, because rather than seeing the secant-tangent case as a degenerate form, the book presents it as a third, nameless theorem). Impenetrable, tear-evoking text. Real world examples that all rely on the two secants being perpendicular.

The truth is, it’s not all that common that you’ll need to determine the curvature of a circle based on a section. It’s not particularly high on the “real world uses” for geometry.

That’s not to say it’s entirely spurious, though. The proof relies on similar triangles. In fact, it’s a very interesting manifestation of the ratio relationship of similar triangles, because it’s a place where we don’t necessarily expect them. Such a connection reinforces the tie between triangles and circles, which is crucial and too often glossed over.

And for me, that’s where the beauty is: Here is something involving a point and a circle and some line segments, and as a result of the ratios of similar triangles, it turns out there’s this cool connection between the segment lengths that’s not obvious just by looking at the picture.

This is where the art of mathematics is, in finding these patterns. The fact that we can find a pattern like this means we can find other patterns, and while this pattern might not be useful unless you’re an architect designing a domed stadium or an archaeologist reconstructing a piece of pottery, it means that there are other patterns out there, waiting to be found. Cool patterns, awesome patterns.

If only we didn’t strip them of their beauty by burying them in dry text and strained real-world applications.