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Ptolemy's inequality, the Glossary

In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space.^[1]

35 relations: Absolute value, Astronomer, CAT(k) space, Circle, Collinearity, Concyclic points, Convex polygon, Cross-ratio, Cycle graph, Euclidean distance, Euclidean geometry, Euclidean plane, Euclidean space, Greece in the Roman era, Hadamard space, Howard Eves, Induced path, Inner product space, Inversive geometry, Mathematician, Metric space, Norm (mathematics), Normed vector space, Parallelogram law, Polarization identity, Projective line, Ptolemaic graph, Ptolemy, Ptolemy's theorem, Quadrilateral, Riemannian manifold, Shortest path problem, Triangle inequality, Vertex (geometry), Vertex (graph theory).
Geometric inequalities
Ptolemy

Absolute value

In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.

See Ptolemy's inequality and Absolute value

Astronomer

An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth.

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CAT(k) space

In mathematics, a \mathbf(k) space, where k is a real number, is a specific type of metric space.

See Ptolemy's inequality and CAT(k) space

Circle

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.

See Ptolemy's inequality and Circle

Collinearity

In geometry, collinearity of a set of points is the property of their lying on a single line.

See Ptolemy's inequality and Collinearity

Concyclic points

In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle.

See Ptolemy's inequality and Concyclic points

Convex polygon

In geometry, a convex polygon is a polygon that is the boundary of a convex set.

See Ptolemy's inequality and Convex polygon

Cross-ratio

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line.

See Ptolemy's inequality and Cross-ratio

In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain.

See Ptolemy's inequality and Cycle graph

Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them.

See Ptolemy's inequality and Euclidean distance

Euclidean geometry

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.

See Ptolemy's inequality and Euclidean geometry

Euclidean plane

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted \textbf^2 or \mathbb^2.

See Ptolemy's inequality and Euclidean plane

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

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Greece in the Roman era

Greece in the Roman era (Greek: Έλλάς, Latin: Graecia) describes the Roman conquest of the territory of the modern nation-state of Greece as well as that of the Greek people and the areas they inhabited and ruled historically.

See Ptolemy's inequality and Greece in the Roman era

Hadamard space

In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space.

See Ptolemy's inequality and Hadamard space

Howard Eves

Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics.

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Induced path

In the mathematical area of graph theory, an induced path in an undirected graph is a path that is an induced subgraph of.

See Ptolemy's inequality and Induced path

Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

See Ptolemy's inequality and Inner product space

Inversive geometry

In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves.

See Ptolemy's inequality and Inversive geometry

Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.

See Ptolemy's inequality and Mathematician

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Ptolemy's inequality and Metric space

Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.

See Ptolemy's inequality and Norm (mathematics)

Normed vector space

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined.

See Ptolemy's inequality and Normed vector space

Parallelogram law

In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry.

See Ptolemy's inequality and Parallelogram law

Polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.

See Ptolemy's inequality and Polarization identity

Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

See Ptolemy's inequality and Projective line

Ptolemaic graph

In graph theory, a Ptolemaic graph is an undirected graph whose shortest path distances obey Ptolemy's inequality, which in turn was named after the Greek astronomer and mathematician Ptolemy. Ptolemy's inequality and Ptolemaic graph are Ptolemy.

See Ptolemy's inequality and Ptolemaic graph

Ptolemy

Claudius Ptolemy (Πτολεμαῖος,; Claudius Ptolemaeus; AD) was an Alexandrian mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine, Islamic, and Western European science.

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Ptolemy's theorem

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). Ptolemy's inequality and Ptolemy's theorem are Ptolemy.

See Ptolemy's inequality and Ptolemy's theorem

Quadrilateral

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices).

See Ptolemy's inequality and Quadrilateral

Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.

See Ptolemy's inequality and Riemannian manifold

Shortest path problem

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.

See Ptolemy's inequality and Shortest path problem

Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. Ptolemy's inequality and triangle inequality are geometric inequalities.

See Ptolemy's inequality and Triangle inequality

Vertex (geometry)

In geometry, a vertex (vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect.

See Ptolemy's inequality and Vertex (geometry)

Vertex (graph theory)

In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

See Ptolemy's inequality and Vertex (graph theory)

References

[1] https://en.wikipedia.org/wiki/Ptolemy's_inequality

Ptolemy's inequality, the Glossary

Table of Contents

See also

Geometric inequalities

Ptolemy

References