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

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

From this,

In other words, if are four points in the plane, , or

**3.** Similarly we have

from which

**4.** Finally, we have the equality

from which

where is the centroid of (i.e. the vector sum is ).

See [2] for more details of special cases of these inequalities.

#### References

[1] 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.

[2] T. Andreescu and D. Andrica, Proving some geometric inequalities by using complex numbers, Educatia Matematica Vol. 1, Nr. 2 (2005), pp. 19–26.

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