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Ptolemy's Theorem in Inversive Geometry
If A,B,C,D are any four points, then |(B,C,A,D)| + |(B,A,C,D)| ≥ 1, with
equality if and only if A,B,C,D lie in this order on an i-line.
Proof
We choose a Mobius transformation t which maps D to ∞.
Suppose that it maps A,B,C to A',B',C', respectively. Then
(B,C,A,D) = t(B,C,A,D) = (B',C',A',∞) = B'A'/C'A' = A'B'/A'C', and
(B,A,C,D) = t(B,A,C,D) = (B',A',C',∞) = B'C'/A'C'.
Now, by the Triangle Theorem of Similarity Geometry,
|A'B'|/|A'C'| + |B'C'|/|A'C'| ≥ 1 with equality if and only if
A',B',C' lie in this order on a line L. This is equivalent to
saying that A',B',C',∞ lie in this order on the extended line L+.
Applying t-1, this is equivalent to the statement that A,B,C,D
lie in this order on the i-line t-1(L+).
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