# Ptolemy's Equality/Inequlity via Inversion

### What is this about?

19 November 2013, Created with GeoGebra

### Ptolemy's Theorem

For four distinct coplanar points $A,$ $B,$ $C,$ and $D,$

$AB\cdot CD+AD\cdot BC\ge AC\cdot BD.$

The equality only reached when the four points are concyclic.

### Hint

There are several known proofs of this well known and useful result; the applet above was intended to suggest that inversion might help furnish another proof.

### Proof

### Ptolemy's Theorem

- Ptolemy's Theorem
- Sine, Cosine, and Ptolemy's Theorem
- Useful Identities Among Complex Numbers
- Ptolemy on Hinges
- Thébault's Problem III
- Van Schooten's and Pompeiu's Theorems
- Ptolemy by Inversion
- Brahmagupta-Mahavira Identities
- Casey's Theorem
- Three Points Casey's Theorem
- Ptolemy via Cross-Ratio
- Ptolemy Theorem - Proof Without Word
- Carnot's Theorem from Ptolemy's Theorem

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