Midpoints of a quadrilateral form a parallelogram

February 2, 2012

Take any quadrilateral, like this one

then mark the midpoints, and connect them up.

It sure looks like we’ve built a parallelogram, doesn’t it? The amazing fact here is that no matter what quadrilateral you start with, you always get a parallelogram when you connect the midpoints.

This is the kind of result that seems both random and astonishing. You have to draw a few quadrilaterals just to convince yourself that it even seems to hold. How do you go about proving it in general?

Some students asked me why this was true the other day. I had totally forgotten how to approach the problem, so I got the chance to play around with it fresh. I had two ideas of how to start. The first was to draw another line in the drawing and see if that helped.

Doesn’t it look like the blue line is parallel to the orange lines above and below it? If that were true, that would give us a powerful way forward. It also presages my second idea: try connecting the midpoints of a triangle rather than a quadrilateral.

Here’s what it looks like for an arbitrary triangle.

It sure looks like connecting those midpoints creates four congruent triangles, doesn’t it? In fact, that’s not too hard to prove. Once we know that, we can see that any pair of touching triangles forms a parallelogram. That means that we have the two blue lines below are parallel.

So we can conclude:
Lemma. The blue lines above are parallel.

Theorem. The orange shape above is a parallelogram.
Proof. Draw in that blue line again.

We have the same situation as in the triangle picture from above! Can you see it?
Let’s erase the bottom half of the picture, and make the lines that are parallel the same color:

See that the blue lines are parallel? The top line connects the midpoints of a triangle, so we can apply our lemma!

But the same holds true for the bottom line and the middle line as well! So all the blue lines below must be parallel.

The same holds true for the orange lines, by the same argument.

So the quadrilateral is a parallelogram after all!

I found this quite a pretty line of argument: drawing in the lines from opposite corners turns the unfathomable into the (hopefully) obvious. That resolution from confusion to clarity is, for me, one of the greatest joys of doing math.

The next question is whether we can break the result by pushing back on the initial setup. Does our result hold, for example, when the quadrilateral isn’t convex?

Looks like it will still hold. I’ll leave that one to you.

Here are a few more questions to consider:

  1. How does the area of the parallelogram you get by connecting the midpoints of the quadrilateral relate to the original quadrilateral?
  2. There is a hexagon where, when you connect the midpoints of its sides, you get a hexagon with a larger area than you started with. Can you find a hexagon with this property?
  3. Can you find a hexagon such that, when you connect the midpoints of its sides, you get a quadrilateral?

 

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