Evelyn Lamb: Welcome to My Favorite Theorem. I’m Evelyn Lamb, a freelance math and sci-ence writer in Salt Lake City. And this is my cohost.
Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn?
EL: Pretty good. It’s hot here, but it gets cool enough at night that it’s survivable. It’s not too bad.
KK: It’s just hot here. It’s awful.
EL: Yeah, there’s really something about that dry heat. I lived in Houston for a while. It’s differ-ent here. So on each episode we invite someone on to tell us about their favorite theorem, and today we’re delighted to have Patrick Honner. Hey! Can you tell us a little bit about yourself?
Patrick Honner: Hi I’m happy to be here. Great to see you, Evelyn and Kevin. I’m in Brooklyn. It’s hot and muggy here. It’s never survivable in New York. I’ve got that going for me. I’m really excited to be here. I’m a high school math teacher. I teach at Brooklyn Technical High School. I studied math long ago, and I’m excited to talk about my favorite theorem today.
KK: So what do you have for us?
PH: In thinking about the prompt of what my favorite theorem was, I guess I came to thinking about it from the perspective of a teacher, of course, because that’s what I’ve been doing for the last almost 20 years. So I was thinking about the kinds of theorems I like to teach, that are fun, that I think are really engaging, that are essential to the courses that i teach. A couple came to mind. I teach calculus occasionally, and I think the intermediate value theorem is probably my favorite theorem in calculus. I feel like the mean value theorem gets all the love in calculus. Eve-ryone thinks that’s the most important, but I really like the intermediate value theorem. I really love De Moivre’s theorem as a connection between complex numbers and geometry and alge-bra, and a little bit of group theory in there. But what really stuck out when thinking about what my favorite theorem is was Varignon’s theorem.
KK: I had to look this up.
PH: Well I think a lot of people, they know it when you show it to them, but they don’t know the name of it. That’s also part of why I like it. The name is sort of exotic sounding. It transports them to France somehow.
KK: Varignon’s theorem is a theorem of Euclidean geometry. It’s not that deep or powerful or exciting, but there’s just something about the way you can interact with it and play with it in class, and the way you prove it and the different directions it goes that really makes it one of my favorite theorems.
KK: Now we’re intrigued.
EL: Yeah. What is this theorem?
PH: Imagine, so Varignon’s theorem is a theorem about quadrilaterals. If you imagine a quadrilateral in the plane, you’ve got the four sides. If you construct the midpoints of each of the four sides, and then connect them in a consistent orientation, so clockwise or counterclockwise, then you will get another quadrilateral. You start with the four sides, take the midpoints and connect them. Now you’ve got another quadrilateral. So if you start with a square, you can imagine those mid-points appearing, and you connect them, then that new quadrilateral would be a square. So you have a square inside of a square. This is a picture I think a lot of people can see.
If you started with a rectangle and you constructed those midpoints, if the rectangle were a non-square rectangle, so longer than it was wide, you can think about it for a moment and maybe draw it, and you’d see a rhombus. Sort of a long, skinny rhombus, depending on the nature of the rectangle. Varignon’s theorem says that regardless of whatever quadrilateral you start with, the quadrilateral you form from those midpoints will be a parallelogram. And I just think that this is so cool.
KK: It’s always a parallelogram.
EL: Yeah, that’s really surprising. By every quadrilateral, do you mean only convex ones, is this for all quadrilaterals?
PH: That’s part of the reason why it’s so much fun to play around with this theorem. It’s true for every quadrilateral, and in fact in some ways, it’s true even for things that aren’t quadrilaterals. In some ways it’s this continual intuition-breaking process with kids when you’re playing around with them. The way you can engage a class with this is you can just tell every student to draw their own quadrilateral and then perform this procedure where they construct the midpoints and connect them. Then you can tell them, ‘Look around. What do you see?’ The first thing the kids see is that everybody drew a square and everybody has a square inscribed, right?
So this is a nice opportunity to confront kids about their mathematical prejudices. Like if you ask them to draw a quadrilateral, they draw a square. If you ask them to draw a triangle, they draw an equilateral triangle. But then there will always be a couple of kids who drew something a little bit more interesting. You can get kids thinking about what all of those things have in common and start looking for a conjecture. You can kind of push the and prod them to maybe do some different things. So maybe on the next interaction of this activity, we’ll get some rectangles or some arbitrary, some non-special quadrilaterals. Even after a couple rounds of this, you’ll still see that almost all the quadrilaterals drawn are convex. Then you can start pushing the kids to see if they understand that there’s another way to draw a quadrilateral that might pose a problem for Varignon’s theorem. It’s so cool that when you get to that convex one, kids never believe that it’ll still form a parallelogram.
EL: In the non-convex one.
PH: That’s right, the concave one, the non-convex. I always get the two words mixed up. Maybe that’s why the kids are so confused. Yeah, the kids will never believe in the non-convex case that it’ll still form a parallelogram. Wow, I can’t believe that.
KK: It seems like, I looked this up, even if the thing isn’t really a quadrilateral, if you take four points in the plane and draw two triangles that meet where the lines cross, it still works, right?
PH: Yeah. There’s yet another level to go with this. Now you’ve got the kids like, wait, so for concave, this works? It’s kind of mind-blowing. Then you can start messing around with their idea of what a quadrilateral actually is. If you show them, well, what if I drew a complex quadri-lateral. I don’t use that terminology right away, but just this idea of connecting the vertices in such a way that two sides appear to cross. It can’t possibly work there, can it? The kids don’t know what to think at this point. They think something weird is going around. Amazingly, even if the quadrilateral crosses itself like that, as long as it’s the non-degenerate case, the four points will still make a parallelogram. It’s really remarkable.
KK: Is there a slick proof of this, or is it one of these crazy things, and you have to construct and construct and construct, and before you know it you’ve lost track of what you’re doing?
PH: No, that’s another reason why this is such a great high school activity. The proof is really accessible. In fact there are several proofs. But before we talk about my favorite proof of my favorite there, there’s another case, another level you can go with Varignon’s theorem. Often I’ll leave this with students as something to think about, a homework problem or something like that. Varignon’s theorem actually works even if the four points don’t form a quadrilateral, so if the four points aren’t coplanar, say. This process of connecting the midpoints will still form a parallelogram. It’s amazing just that the four points are coplanar. You wouldn’t necessarily expect that the four midpoints would be in the same plane if the four starting points aren’t in the same plane. Moreover, those four points form a parallelogram. It’s such an amazing thing.
EL: What is your favorite proof, then?
PH: My favorite proof of Varignon’s theorem is something that connects to a couple of key ideas that we routinely explore in high school geometry. The first is one of the first important theorems about triangles that we prove, that’s simple but has some power. It’s that if you connect the mid-points of two sides of a triangle, that line segment is parallel to the third side. And it’s also half the length. But the parallelism is important.
The other idea, and I think this is one of the most important ideas that i try to emphasize with students across courses, is the idea of transitivity, of equality or congruence, or in this case par-allelism. The nice proof of Varignon’s theorem is that you imagine all the quadrilaterals and midpoints. And you draw one diagonal. You just think about one diagonal. Now if you cover up half of the quadrilateral, you’ve got a triangle. The line segment connecting those two midpoints is parallel to that diagonal because that’s just that triangle theorem. Now if you cover up the other half of the quadrilateral, you have a second triangle. And that segment is parallel to the diagonal. So both of those line segments are parallel to that diagonal, and therefore by transitivity, they’re parallel to each other, and now you have that the two opposite sides are parallel. And the exact same argument works for the other sides using the other diagonal.
KK: I like that. My first instinct would be to do some sort of vector analysis. You realize all the sides as vectors and then try to add them up and show that they’re parallel or something.
PH: Yeah, and in some of the courses i teach, I do some work with vectors, and this is definitely something we do. We explore that proof using vectors, or coordinate geometry. Maybe later in the year we’ll do some work with coordinate geometry. We can prove it that way too.
EL: Yeah, I think I would immediately go to coordinates. Of course, I would have assumed they were coplanar in the first place. If you tell me it’s a quadrilateral, yeah, it’s going to be there in the plane and not in 3-space.
PH: I love coordinate geometry, and I definitely have that instinct to run to coordinates when I want to prove something. One of those things you have to be careful of in the high school class is making sure they understand all the assumptions that underly the use of coordinates, and un-derstanding the nature of an arbitrary figure. Going back to one of the first things I said, if you ask kids to draw a quadrilateral, they’re going to draw a square, or if you ask them to draw an arbitrary quadrilateral, they’re often going to draw a square or rectangle. If you ask them to draw an arbitrary quadrilateral in the plane, they might make assumptions about where those coordi-nates are likely to be.
KK: Your students are lucky to have you.
PH: That’s what I tell them!
KK: Really, to give this much thought to something like this and show all these different per-spectives and how you might come at it in all these different ways, my high school geometry class, I mean I had a fine teacher, but we never saw anything with this kind of sophistication at all.
PH: It’s fun. I would like to present it as if I sat around and thought deeply about it and had this really thoughtful approach to it, but it just kind of happened. I think, again, that’s why this is one of my favorite theorems. You can just put this in front of students and have them play and just run with this. It’ll just go in so many different directions.
EL: So what have you chosen to pair with this theorem? What do you enjoy experiencing along with the glory of this theorem?
PH: This was a tricky one. I feel like when I think of Varignon’s theorem, really focusing on the name, it really transports me to France. I feel like it’s a hearty stew, like boeuf Varignon or something like that. I think you need some crusty bread and a glass of red wine with Varignon’s theorem]. Not my students.
EL: Crusty bread and grape juice for them. Yeah, I just got back from living in France for six months, and actually I didn’t have any boeuf bourgignon, or Varignon, while I was there, but I did enjoy quite a few things with crusty bread and a glass of red wine. I highly recommend it.
KK: This has been great fun.
PH: Yeah, I’ve enjoyed this. You seem to enjoy talking about this more than my students, so this was great for me.
KK: It helps to be talking to a couple of mathematicians, yeah.
EL: So, we like to let guests plug websites or anything. So would you like to tell people about your blog or any things you’re involved in that you’d like to share?
PH: Yeah, sure. I blog, less frequently now than I used to, but still pretty regularly. I blog at mrhonner.com. You can generally find out about what I’m doing at my personal website, pat-rickhonner.com. I’m pretty active on Twitter, @mrhonner.
KK: Lots of good stuff on Patrick’s blog, especially after the Regents exams. You have a lot to say.
PH: Not everybody thinks it’s good stuff. I’m glad some people do.
KK: I don’t live in New York. It’s fine with me.
EL: Yeah, he has a series kind of taking apart some of the worst questions on the New York Regents exams for math. It can be a little frustrating.
PH: We just wrapped up Regents season here. Let’s just say there are some posts in the works about what we’re facing. You know, I enjoy it. It always sparks interesting mathematical conver-sations. My goal is just to raise awareness about the toll of these tests and how sometimes it seems like not enough attention is given to making sure these tests are of high quality and are valid.
KK: I don’t think it’s just a problem in New York, either.
PH: It is not just a problem in New York.
KK: Well thanks for joining us, Patrick. This was really great. I learned something today.
EL: Yeah, me too.
PH: It was my pleasure. Thanks for having me. Thanks for giving me an opportunity to think about my favorite theorem and come on and talk about it. And maybe Varignon’s theorem will appear in a couple more geometry classes next year because of it.
KK: Let’s hope.
EL: Yeah, I hope so.
KK: Take care.
PH: Thanks. Bye.