The following inequalities all follow from the elementary triangle inequality for complex numbers:
Equality holds if and only if are collinear or in other words is a real number when .
1. Ptolemy’s inequality: we start with the equality
Hence if are four points in the plane,
which is Ptolemy’s inequality. Equality holds if and only if
. The ratio is a complex number with argument while the ratio has argument . Hence for the product of these ratios to be real means is a multiple of . In other words, the points lie on a circle.
2. We start with the equality
In other words, if are four points in the plane, , or
3. Similarly we have
4. Finally, we have the equality
where is the centroid of (i.e. the vector sum is ).
See  for more details of special cases of these inequalities.
 A. Bogomolny, Complex Numbers and Geometry from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/arithmetic/algebra/ComplexNumbersGeometry.shtml#cycl, Accessed 22 April 2013.
 T. Andreescu and D. Andrica, Proving some geometric inequalities by using complex numbers, Educatia Matematica Vol. 1, Nr. 2 (2005), pp. 19–26.