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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.
In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey. Formulation of the theorem [ edit ]
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
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The Planisphaerium is a work by Ptolemy. The title can be translated as "celestial plane" or "star chart". In this work Ptolemy explored the mathematics of mapping figures inscribed in the celestial sphere onto a plane by what is now known as stereographic projection. This method of projection preserves the properties of circles.
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