Things continued difficult today, but I felt a bit better after dinner. I watched four videos from the Numberphile channel, notably one about proving a geometric relationship called Ptolemy’s Theorem by a technique called inversion of the plane. It was a great video, but quite involved. I would not recommend it to those without a strong interest in math.
When I am feeling better, I would like to have a go at proving Ptolemy’s Theorem by another method.1 The lecturer in the video said that it can be done using plane geometry, provided you are clever, and can also be done by crunching trig ratios, something I have some experience in. I’d also like to prove, by some other method, a fact that can be proven using Ptolemy’s Theorem: that in the figure shown below, the distance from $D$ to $B$ is the sum of the distances from $D$ to $A$ and $C$.
A cool thing about Ptolemy’s Theorem is that the Pythagorean Theorem falls right out of it. In fact, you could say that the Pythagorean Theorem is a special case of Ptolemy’s. This means that Ptolemy’s Theorem must only be true if you assume the Parallel Postulate. I wonder if that is the case for the proposition above.
- To do. âŠī¸
How cool. You’re really teaching me some geometry.
I really like you’re approach, too. Exploring different proofs does at least two things:
(1) It helps understand why the theorem is true, and how to situate it in the wider mathematical world.
(2) Maybe even more importantly, it helps you own the theorem. Even if your proof is less miraculous than one using a technique like inversion, it still lets you say, OK. I understand the theorem. I could have priced it myself. And who knows, maybe your proof will be that miraculous.
Your drive to deep understanding is so powerful and beautiful. Your blog is giving me so much happiness!
It gives me happiness, too, when I am in a state to feel it. Your comments also give me happiness.
Cool to try to prove it in a different way. You go, girl!
I go, at least. đ