Ptolemy's Equality/Inequlity via Inversion

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19 November 2013, Created with GeoGebra

Ptolemy's Theorem

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

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

  2. 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

A proof is available on a separate page. Below are just two illustrations:

Ptolemy's Theorem

  1. Ptolemy's Theorem
  2. Sine, Cosine, and Ptolemy's Theorem
  3. Useful Identities Among Complex Numbers
  4. Ptolemy on Hinges
  5. Thébault's Problem III
  6. Van Schooten's and Pompeiu's Theorems
  7. Ptolemy by Inversion
  8. Brahmagupta-Mahavira Identities
  9. Casey's Theorem
  10. Three Points Casey's Theorem
  11. Ptolemy via Cross-Ratio
  12. Ptolemy Theorem - Proof Without Word
  13. Carnot's Theorem from Ptolemy's Theorem

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