This is a cool theorem from multivariable calculus that my professor at Western Michigan University, Steve Mackey, showed us during lecture one day early in the semester.

** Theorem**: Let \(a\), \(b\), \(c\), and \(d\) be any four points in \(\mathbb{R}^{3}\). Let \(M\), \(N\), \(P\), and \(Q\) be the midpoints between the adjacent points. Then vector

**MQ**must always be identical to vector

**NP**.

I have inserted a picture to help visualize it. I made the picture in PowerPoint, so it’s about as accurate as a hand-drawn picture.

It looks like the points are all in the same plane, but that’s just because it’s easier to draw them that way. They can be *any four points* in three-dimensional space.

** Proof**: Start with vector

**MQ**. By the basic rules of vector addition,

MQ=MA+AQ

= 1/2BA+ 1/2AD

= 1/2 (BA+AD)

= 1/2BD

Now do the same with vector **NP**:

NP=NC+CP

= 1/2 (BC+CD)

= 1/2BD

Thus, **MQ** = **NP** = 1/2 **BD**. ■

The reason this is so cool is because it holds true for *any* four points in \(\mathbb{R}^{3}\), which makes it very unexpected. You can rearrange the names of the points and midpoints so that the vector labeled **MQ** isn’t identical to the vector labeled **NP**, but then two other vectors will be identical. The point is that given any four points in three-dimensional space, some pair of midpoint-connecting vectors will be identical.

After I wrote this post, I realized something that Dr. Mackey didn’t mention (or at least, I didn’t write in my notes): two other vectors must also be identical. Can you show which ones?