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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 ...
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 ...
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 ...
Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces , and other distances than Euclidean have been studied.
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.
U.S. President-elect Donald Trump is making good on his threats to go after the media in court, with several recent lawsuits seeking damages against major publishers over what he describes as ...
Ptolemy's model of astronomy was used as a technical method that could answer questions regarding astrology and predicting planets positions for almost 1,500 years, even though the equant and eccentric were regarded by many later astronomers as violations of pure Aristotelian physics which presumed all motion to be centered on the Earth. It has ...