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[[Image:Ptolemy Inequality.svg|thumb|Four points and their six distances. The points are not co-circular, so Ptolemy's inequality is strict for these points.]]
In [[Euclidean geometry]], '''Ptolemy's inequality''' relates the six [[Euclidean distance|distances]] determined by four points in the [[Euclidean plane|plane]] or in a higher-dimensional space. It states that, for any four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|D}}, the following [[Inequality (mathematics)|inequality]] holds:
:<math>\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge \overline{AC}\cdot \overline{BD}.</math>
It is named after the Greek [[Astronomy:Astronomer|astronomer]] and [[Biography:Mathematician|mathematician]] [[Biography:Ptolemy|Ptolemy]].

The four points can be ordered in any of three distinct ways (counting reversals as not distinct) to form three different [[Quadrilateral|quadrilateral]]s, for each of which the sum of the products of opposite sides is at least as large as the product of the diagonals. Thus, the three product terms in the inequality can be additively permuted to put any one of them on the right side of the inequality, so the three products of opposite sides or of diagonals of any one of the quadrilaterals must obey the [[Triangle inequality|triangle inequality]].<ref name="s40">{{citation
 | doi = 10.2307/1968849
 | journal = Annals of Mathematics
 | mr = 0002903
 | pages = 715–726
 | series = Second Series
 | title = On metric arcs of vanishing Menger curvature
 | volume = 41
 | year = 1940| issue = 4 | jstor = 1968849 }}.</ref>

As a special case, [[Ptolemy's theorem]] states that the inequality becomes an equality when the four points lie in cyclic order on a [[Circle|circle]].
The other case of equality occurs when the four points are [[Collinearity|collinear]] in order. The inequality does not generalize from [[Euclidean space]]s to arbitrary metric spaces. The spaces where it remains valid are called the ''Ptolemaic spaces''; they include the [[Inner product space|inner product space]]s, [[Hadamard space]]s, and shortest path distances on [[Ptolemaic graph]]s.

==Assumptions and derivation==
Ptolemy's inequality is often stated for a special case, in which the four points are the [[Vertex (geometry)|vertices]] of a [[Convex polygon|convex]] [[Quadrilateral|quadrilateral]], given in cyclic order.<ref>{{citation|title=The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities|series=MAA problem books|first=J. Michael|last=Steele|publisher=Cambridge University Press|year=2004|isbn=9780521546775|contribution-url=https://books.google.com/books?id=7GDyRMrlgDsC&pg=PA69|page=69|contribution=Exercise 4.6 (Ptolemy's Inequality)}}.</ref><ref>{{citation|title=When Less is More: Visualizing Basic Inequalities|volume=36|series=Dolciani Mathematical Expositions|first1=Claudi|last1=Alsina|first2=Roger B.|last2=Nelsen|publisher=Mathematical Association of America|year=2009|isbn=9780883853429|pages=82–83|contribution-url=https://books.google.com/books?id=U1ovBsSRNscC&pg=PA82|contribution=6.1 Ptolemy's inequality}}.</ref> However, the theorem applies more generally to any four points; it is not required that the quadrilateral they form be convex, simple, or even planar.

For points in the plane, Ptolemy's inequality can be derived from the [[Triangle inequality|triangle inequality]] by an inversion centered at one of the four points.<ref>{{harvtxt|Apostol|1967}} attributes the inversion-based proof to textbooks by R. A. Johnson (1929) and [[Biography:Howard Eves|Howard Eves]] (1963).</ref><ref name="bmc">{{citation|title=A Decade of the Berkeley Math Circle: The American Experience|volume=1|series=MSRI Mathematical Circles Library|editor1-first=Zvezdelina|editor1-last=Stankova|editor1-link=Zvezdelina Stankova|editor2-first=Tom|editor2-last=Rike|publisher=American Mathematical Society|year=2008|isbn=9780821846834|page=18|contribution-url=https://books.google.com/books?id=vix-AwAAQBAJ&pg=PA18|contribution=Problem 7 (Ptolemy's Inequality)}}.</ref> Alternatively, it can be derived by [[Ptolemy's theorem#Proof using complex numbers|interpreting the four points as complex numbers]], using the complex number identity:
:<math>(A-B)(C-D)+(A-D)(B-C)=(A-C)(B-D)</math>
to construct a triangle whose side lengths are the products of sides of the given quadrilateral, and applying the triangle inequality to this triangle.{{sfn|Apostol|1967}} One can also view the points as belonging to the complex [[Projective line|projective line]], express the inequality in the form that the [[Absolute value|absolute value]]s of two [[Cross-ratio|cross-ratio]]s of the points sum to at least one, and deduce this from the fact that the cross-ratios themselves add to exactly one.<ref>{{citation|title=Geometry: Ancient and Modern|first=John R.|last=Silvester|publisher=Oxford University Press|year=2001|isbn=9780198508250|contribution-url=https://books.google.com/books?id=VtH_QG6scSUC&pg=PA229|page=229|contribution=Proposition 9.10 (Ptolemy's theorem)}}.</ref>

A proof of the inequality for points in three-dimensional space can be reduced to the planar case, by observing that for any non-planar quadrilateral, it is possible to rotate one of the points around the diagonal until the quadrilateral becomes planar, increasing the other diagonal's length and keeping the other five distances constant.{{sfn|Apostol|1967}} In spaces of higher dimension than three, any four points lie in a three-dimensional subspace, and the same three-dimensional proof can be used.

==Four concyclic points==
{{main|Ptolemy's theorem}}
For four [[Concyclic points|points in order around a circle]], Ptolemy's inequality becomes an equality, known as [[Ptolemy's theorem]]:
:<math>\overline{AB}\cdot \overline{CD}+\overline{AD}\cdot\overline{BC} = \overline{AC}\cdot \overline{BD}.</math>
In the inversion-based proof of Ptolemy's inequality, transforming four co-circular points by an inversion centered at one of them causes the other three to become collinear, so the triangle equality for these three points (from which Ptolemy's inequality may be derived) also becomes an equality.<ref name="bmc"/> For any other four points, Ptolemy's inequality is strict.

==In three dimensions==
Four non-coplanar points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|D}} in 3D form a tetrahedron. In this case, the strict inequality holds:
<math>\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} > \overline{AC}\cdot \overline{BD}</math>.<ref>{{cite journal |last1=Zhu |first1=Hanlin |title=68.25 A Tetrahedron Inequality |journal=The Mathematical Gazette |date=1984 |volume=68 |issue=445 |pages=200–202 |doi=10.2307/3616345 |url=https://www.jstor.org/stable/3616345 |issn=0025-5572}}</ref>

==In general metric spaces==
[[File:Circle graph C4.svg|thumb|upright=0.75|A [[Cycle graph|cycle graph]] in which the distances disobey Ptolemy's inequality]]
Ptolemy's inequality holds more generally in any [[Inner product space|inner product space]],<ref name="s40"/><ref name="nls"/> and whenever it is true for a real [[Normed vector space|normed vector space]], that space must be an inner product space.<ref name="nls">{{citation|title=Introduction to the Analysis of Normed Linear Spaces|volume=13|series=Australian Mathematical Society lecture series|first=J. R.|last=Giles|publisher=Cambridge University Press|year=2000|isbn=9780521653756|page=47|contribution=Exercise 12|contribution-url=https://books.google.com/books?id=VVeV6EKimjQC&pg=PA47}}.</ref><ref>{{citation
 | doi = 10.2307/2031742
 | journal = Proceedings of the American Mathematical Society
 | mr = 0052035
 | pages = 961–964
 | title = A remark on M. M. Day's characterization of inner-product spaces and a conjecture of L. M. Blumenthal
 | volume = 3
 | year = 1952| issue = 6 | jstor = 2031742 }}.</ref>

For other types of [[Metric space|metric space]], the inequality may or may not be valid. A space in which it holds is called ''Ptolemaic''. For instance, consider the four-vertex [[Cycle graph|cycle graph]], shown in the figure, with all edge lengths equal to 1. The sum of the products of opposite sides is 2. However, diagonally opposite [[Vertex (graph theory)|vertices]] are at distance 2 from each other, so the product of the diagonals is 4, bigger than the sum of products of sides. Therefore, the shortest path distances in this graph are not Ptolemaic. The graphs in which the distances obey Ptolemy's inequality are called the [[Ptolemaic graph]]s and have a restricted structure compared to arbitrary graphs; in particular, they disallow induced cycles of length greater than three, such as the one shown.<ref>{{citation
 | last = Howorka | first = Edward
 | doi = 10.1002/jgt.3190050314
 | issue = 3
 | journal = Journal of Graph Theory
 | mr = 625074
 | pages = 323–331
 | title = A characterization of Ptolemaic graphs
 | volume = 5
 | year = 1981}}.</ref>

The Ptolemaic spaces include all [[CAT(k) space|CAT(0) spaces]] and in particular all [[Hadamard space]]s. If a complete [[Riemannian manifold]] is Ptolemaic, it is necessarily a Hadamard space.<ref>{{citation
 | last1 = Buckley | first1 = S. M.
 | last2 = Falk | first2 = K.
 | last3 = Wraith | first3 = D. J.
 | doi = 10.1017/S0017089509004984
 | issue = 2
 | journal = Glasgow Mathematical Journal
 | mr = 2500753
 | pages = 301–314
 | title = Ptolemaic spaces and CAT(0)
 | volume = 51
 | year = 2009| doi-access = free
 }}.</ref>

==Inner product spaces==

Suppose that <math>\|\cdot\|</math> is a [[Norm (mathematics)|norm]] on a vector space <math>X.</math> Then this norm satisfies Ptolemy's inequality:
<math display=block>\|x - y\| \, \|z\| ~+~ \|y - z\| \, \|x\| ~\geq~ \|x - z\| \, \|y\| \qquad \text{ for all vectors } x, y, z.</math>
if and only if there exists an inner product <math>\langle \cdot, \cdot \rangle</math> on <math>X</math> such that <math>\|x\|^2 = \langle x,\ x\rangle</math> for all vectors <math>x \in X.</math><ref>{{Cite journal|last=Apostol|first=Tom M. | date=1967 |title=Ptolemy's Inequality and the Chordal Metric |url=https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1967.11975804 |journal=Mathematics Magazine |volume=40 |issue=5 |pages=233–235 |language=en |doi=10.2307/2688275 |jstor=2688275 | mr = 0225213}}</ref> Another necessary and sufficient condition for there to exist such an inner product is for the norm to satisfy the [[Parallelogram law|parallelogram law]]:
<math display=block>\|x+y\|^2 ~+~ \|x-y\|^2 ~=~ 2\|x\|^2 + 2\|y\|^2 \qquad \text{ for all vectors } x, y.</math>
If this is the case then this inner product will be unique and it can be defined in terms of the norm by using the [[Polarization identity|polarization identity]].

==See also==

* {{annotated link|Parallelogram law}}
* {{annotated link|Polarization identity}}
* {{annotated link|Biography:Ptolemy}}
* {{annotated link|Ptolemy's table of chords}}
* {{annotated link|Ptolemy's theorem}}

==References==
{{reflist|30em}}

{{Ancient Greek astronomy}}
{{Ancient Greek mathematics}}

[[Category:Geometric inequalities]]

{{Sourceattribution|Ptolemy's inequality}}