Lessons taught; Lessons learnt

Following on from the previous post, here is an applet which dynamically generates a pentagon, given the midpoints. Have a play around, and then continue reading for some follow on questions.

Geogebra applet will be here if you enable Java.

(Source: pentagon_given_midpoints)

This does demonstrate that we can find a pentagon with any given midpoints, but is it definitive evidence? No, for a very important reason: with any dynamic construction, we will only be able to test a finite number of points… a proof needs to be true for every possible combination of midpoints.

Also, although the applet gives a single pentagon for each combination of midpoints, that doesn’t mean that there aren’t others… it just means that I’ve only constructed one solution in this applet.

Just as with the previous post, on quadrilaterals, it is in fact easy to show using vectors that there is a unique solution for any possible combination of midpoints. In addition, the vertices of the pentagon have a very nice vector representation in terms of the midpoints… but I’ll leave finding that representation to you!

Here are some interesting follow up questions about this situation, which you might want to think about, possibly with the help of the applet:

First, suppose we keep four of the points fixed, and alter the position of the fifth midpoint. Can we find a simple description of what happens to the vertices of the pentagon as we move the midpoint around?

Second, a lot of the midpoint configurations give results which are not ‘traditional’ pentagons. Like these, for example:

Can we find a restriction on the position or the order of the midpoints which rules out these shapes which have edges intersecting at points other than the vertices?

Third, it’s fairly simple to generate ‘degenerate’ pentagons which are really quadrilaterals:

Can we explain which configurations of midpoints lead to quadrilaterals?

Fourth, is it possible to generate degenerate pentagons which are really triangles in the same way?

I’m sure we can come up with other questions (and I’d like to hear about any interesting results people find), but there’s enough here to keep me going for the moment!

Related Posts (automatically generated)

1. Midpoints of quadrilaterals
2. Midpoints (3): A return to quadrilaterals
3. Visualisation: Rotating polygons