Cool theorem about midpoints and parallel vectors from multivariable calculus

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
= 1/2 BA + 1/2 AD
= 1/2 (BA + AD)
= 1/2 BD

Now do the same with vector NP:

NP = NC + CP
= 1/2 (BC + CD)
= 1/2 BD

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?

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