Ptolemy's theorem

Students must know about cyclic quadrilaterals before learning about Ptolemy's theorem.

In this topic, students will learn what Ptolemy's theorem is, and why it's true. Here's the most elegant proof. Next, have students attempt to prove Ptolemy's theorem another way, as seen here. Provide the hint on that page, then give your students some time to think. If students fail to find the proof for themselves, that's okay, you can simply give them the proof. Before attempting to understand these proofs, students must understand similar triangles, and know that angles standing on the same arc are equal. There's another proof here, involving circle inversion, but I think it would be too difficult for students at this stage. Here are two interesting problems you can give your students on Ptolemy's theorem:

Prove \(b + c = d\)

Prove \(\dfrac{a}{b} = \dfrac{1 + \sqrt{5}}{2}\)

Here are their solutions.

Challenge problem:

An ant is standing on a regular dodecahedron. When standing on a face, the ant can see only that face. When standing on an edge, the ant can see only the two faces adjacent to that edge. When standing on a vertex, the ant can see only the three faces adjacent to that vertex. What path can the ant take, such that the ant never walks along a face, the path ends where it begins, and all faces are seen? What if the ant is allowed to walk along a face? Here's the answer.