Ptolemy's Equality/Inequlity via Inversion

What is this about?

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:

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Copyright © 1996-2018 Alexander Bogomolny
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