Ptolemy Theorem:

• webpage: Ptolemy Theorem, on cut-the-knot portal, by A. Bogomolny.
• webpage: A collection of proofs of Ptolemy's Theorem, by ckrao.
• video: Zvezdelina Stankova, A Miraculous Proof (Ptolemy's Theorem) Numberphile.

Some advanced/exotic versions of Ptolemy:

• A. Berenstein, V. Retakh, Noncommutative Marked Surfaces (noncommutative analogue of Ptolemy).
• video: NJ Wildberger, Ptolemy's theorem and generalizations, including Ptolemy over finite fields.

Friezes and SL2-tilings:

• webpage: Friezes. My own collection of links.
• video: Sergei Tabachnikov, Frieze Patterns, Numberphile.
• video: Sergei Tabachnikov, Frieze Patterns, extra, Numberphile.

Hyperbolic Ptolemy:

• Boris Springborn, Hyperbolic polyhedra and discrete uniformization.
• Boris Springborn, The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms.
• A. Felikson, O. Karpenkov, K. Serhiyenko, P. Tumarkin, 3D Farey graph, lambda lengths and SL2-tilings (3D version of Ptolemy).
• Daniel V. Mathews, Spinors and horospheresSpinors and horospheres.

Grassmannians and cluster algebras:

• video: Lauren Williams, The positive Grassmannian, the amplituhedron, and cluster algebras.

Cluster algebras:

• website: Sergey Fomin, Cluster Algebras Portal.
• video: Andrei Zelevinsky, Cluster algebras.

Ptolemy spaces:

• webpage: Anna Felikson, Ptolemy spaces.