Thursday, November 16, 2023

Ratio boxes

I worked up a short new chapter for my Geometry book. It's about a device I'm calling ratio boxes, for want of a better word. When we have similar triangles, we have equal ratios of sides.

An example:

Above we have three similar right triangles, so we write down the sides in order from smallest to largest, and then repeat, going through each triangle in order.

The trick is that any four entries making a rectangle are a valid ratio from this data.

In particular, I'm hoping you may be able to see a quick proof of Pythagoras's Theorem.

There are several more examples. The most complicated is one from Inversive Transformation in a circle. The rule for the transformation is OA times OA' = r^2, where r is the radius of the circle with the solid line.

As we work through the example, you should be able to see how the ratio boxes dramatically simplify the bookkeeping involved in the proof. The chapter is on my Dropbox as a pdf.

The theorem is one of my very favorites.

Tuesday, November 7, 2023

Napoleon's Theorem

Napoleon's Theorem is a theorem some attribute (naturally enough) to Napoleon.

It says that if you take any triangle and draw equilateral triangles on each side, then the incenters of those triangles form a fourth equilateral triangle.

There is a variant in which the new triangles are drawn as reflections of the other ones, that is, inside the original triangle.

There is a terrific vector proof that I diagram here. (I think I got the idea for the proof from Alexander Bogomolny, but I can't find it at the moment. Wonderful site).

Define vectors for paths to and from the incenters based on the following.
Then apply a simple test for the adjacent sides of an equilateral triangle:
The details depend on the definition of the direction of rotation, and the path taken around the putative equilateral triangle. Details in the links below. Here is a variant of the problem:
My write-up is here. Probably the neatest thing is we get the variant basically for free, once the setup is done. I also (finally) got a proof on ProofWiki here as well as the variant (here)