In euclidean geometry, we have the following classical result
Ptolemy's Theorem
If the points A,B,C,D lie in this order on the circumference of a circle, then
ABCD + ADBC = ACBD.
Some texts contain a better result, which hints at a connection with
inversive geometry since it involves an object which is a line or circle,
i.e. an iline.
Strong form of Ptolemy's Theorem
For any four points, ABCD + ADBC ≥ ACBD, with equality
if and only if A,B,C,D lie in this order on a line or circle.
Immediately we see, by dividing through by ACBD, that each result
involves only ratios, so is a theorem of similarity geometry. In fact, we
will show that the theorem really belongs to inversive geometry, and it
can be viewed as a generalisation of the
Triangle Inequality in Similarity Geometry
If A,B,C are any three points then AB/AC + BC/AC ≥ 1, with
equality if and only if A,B,C lie in this order on a line.
This is obtained from the standard euclidean triangle inequality AB + BC ≥ AC,
by dividing through by AC.
To say that A,B,C lie in this order on a line is just another way of saying
that B lies between A and C ( or on the segment AC).


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 iline.
proof
To recover the strong form of the euclidean theorem, we simply observe
that (B,C,A,D) = BACD/CABD and (B,A,C,D) = BCAD/ACBD.

