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Butterfly Theorem: History & Applications

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

The butterfly theorem and its applications have kept mathematicians fascinated for many years. We will take a look at this theorem, explore its history a bit, and discuss its application in various areas.

Butterfly Theorem

Humor me for a moment. Take out a pencil and some paper, and do the following:

  1. Draw a circle and a chord of the circle running from any two points P and Q on the circle.
  2. Let the midpoint of PQ be M. Draw two more chords of the circle, call them RS and TV, that both pass through M.
  3. Lastly, add in the two line segments RV and ST.

Okay, now think of some different types of bugs or insects. Do you see any resemblance between the shape you just created in your drawing and a particular type?


Butterfly1


That's right! It looks like a butterfly!

We have a nice theorem pertaining to the scenario we just drew, and as you can see why, it is called the butterfly theorem.

  • Butterfly Theorem: For any chord PQ of a circle, let M be the midpoint of PQ. If we draw two other chords, RS and VT, through M, draw the line segments RV and ST, and call the intersection points of RV and ST with PQ Y and X, then M is the midpoint of XY.


Butterfly2


That's a bit of a mouthful, but it basically just takes us through the steps we followed in our drawing, and then states that if we call the intersection points of RV and ST with PQ Y and X, then M is not only the midpoint of PQ, but it is also the midpoint of XY. If you think of this in terms of a butterfly, we could think of M as a point on the body of the butterfly, and we see that M is always going to be directly in the center of opposite edges of the butterfly's wings.

Well, that's pretty interesting! Let's take a look at a brief history of this neat little theorem.

Brief History

There are many different proofs of the butterfly theorem, but its original proof is credited to William Wallace. Originally, it was thought that W. G. Horner was responsible for solving this problem in 1815. However, two recent discoveries have made it clear that Wallace's proof came before Horner's proof by about 10 years.

The first of those discoveries is an 1803 publication of The Gentlemen's Mathematical Companion containing a generalization of the problem by Wallace. The second is a correspondence from Sir William Herschel to Wallace in 1805. In the letter, Herschel presents the problem of proving that the two line segments XM and YM in the theorem are equal in length. In turn, he is asking Wallace for a proof of the butterfly theorem.


1805 Correspondence
Butterfly3


As we said, it is now obvious that it was Wallace that first came up with a proof to this theorem, and it's a good thing he did, because today, we can use the butterfly theorem in various applications.

Application

Suppose that two people are out for a hike in a large circular hiking area, and they have strayed off the path. They have a map of the hiking area with the trails and their distances given.


Butterfly4


From the map, they can see that they need to continue hiking to the west to get back to the parking lot. However, because they have gone off-path, they aren't sure how much farther they need to go, and they're getting tired.

Notice that the trails in the hiking area look very similar to the scenario in the butterfly theorem, except that there's not a trail, or line segment, from Y to M, and that is where the couple is now located. Hmmm…well, if we were to sketch in that missing line segment, then by the butterfly theorem it must be the case that M is the midpoint of X and Y. Because of this, it directly follows that XM and YM have the same length. Ah-ha! There's the couple's answer to this conundrum!

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