Equilateral triangle
Ptolemy's Theorem yields as a corollary a pretty theorem regarding an equilateral triangle inscribed in a circle. 展开

In the case of a circle of unit diameter the sides $${\displaystyle S_{1},S_{2},S_{3},S_{4}}$$ of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles $${\displaystyle \theta _{1},\theta _{2},\theta _{3}}$$ and 展开

• Casey's theorem
• Greek mathematics 展开

Visual proof
The animation here shows a visual demonstration of Ptolemy's theorem, based on Derrick & … 展开

The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral ABCD, then 展开

• Proof of Ptolemy's Theorem for Cyclic Quadrilateral
• MathPages – On Ptolemy's Theorem
• Elert, Glenn (1994). "Ptolemy's Table of Chords". E-World.
• Ptolemy's Theorem at cut-the-knot 展开