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Exploring Menelaus’ Theorem

In the last post, we used several theorems we’d developed to arrive at Menelaus’ theorem. However, at the very end Ptolemy simply mentions another version of the theorem, but does not derive it. I simply took his word that it worked, but as that alternative form is used first thing in the next chapter, I want to make sure at the very use, we know how to use it, even if we don’t go through how it’s derived.

First, let’s set up a generic Menelaus configuration on a sphere which is the intersection of the arcs of four great circles:

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Almagest Book I: Menelaus’ Theorem

So far in these preliminary theorems, we’ve looked at some that were based on triangles and some that were based on circles. We’ll be going one step further with this next one and work with spheres. Thus far, we’ve briefly touched on spheres in this post discussing the celestial sphere. If great circles and spherical triangles aren’t familiar to you, I suggest reading over that post.

But since this is the first time we’ve encountered math in 3D if you’ve been following along, I want to build this up more slowly2 and will be trying to add some 3D elements to make the visualization a bit easier.

So let’s get started.

[L]et us draw the following arcs of great circles on a sphere: BE and GD are drawn to meet AB and AG, and cut each other at Z. Let each of them be less than a semi-circle.

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