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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 ...
Download as PDF; Printable version; In other projects ... Shapiro inequality; Stirling's formula (bounds) Differential equations ... Ptolemy's inequality; Pu's ...
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 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 ...
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.
Great Commentary on Ptolemy's Handy Tables. This work partially survives. It originally consisted of 5 books, of which books 1–3 and the beginning of book 4 are extant. It describes how to use Ptolemy's tables and gives details on the reasoning behind the calculations. [1] Little Commentary on Ptolemy's Handy Tables. This work survives complete.
Aristarchus's inequality (after the Greek astronomer and mathematician Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of trigonometry which states that if α and β are acute angles (i.e. between 0 and a right angle) and β < α then
Greek mathematics [a] reached its acme during the Hellenistic and early Roman periods, and much of the work represented by authors such as Euclid (fl. 300 BC), Archimedes (c. 287–212 BC), Apollonius (c. 240–190 BC), Hipparchus (c. 190–120 BC), and Ptolemy (c. 100–170 AD) was of a very advanced level and rarely mastered outside a small ...