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For four points in order around a circle, Ptolemy's inequality becomes an equality, known as Ptolemy's theorem: ¯ ¯ + ¯ ¯ = ¯ ¯. 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 ...
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. The Ptolemaic graphs are exactly the graphs that are both chordal and distance-hereditary ; they include the block graphs [ 1 ] and are a subclass of the ...
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. [25] The Euclidean distance gives Euclidean space the structure of a topological space , the Euclidean topology , with the open balls (subsets of points at less than a given distance from a given point) as its neighborhoods .
Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that + where there is equality if and only if the quadrilateral is cyclic. [24]: p.128–129 This is often called Ptolemy's inequality.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
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1.3 Differential equations. 1.4 Geometry. ... 2 Inequalities particular to physics. 3 See also. ... Ptolemy's inequality; Pu's inequality;