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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). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). [ 1 ]
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 ...
This proof is independent of the Pythagorean theorem, insofar as it is based only on the right-triangle definition of cosine and obtains squared side lengths algebraically. Other proofs typically invoke the Pythagorean theorem explicitly, and are more geometric, treating a cos γ as a label for the length of a certain line segment. [13]
The first of these theorems is the spherical analogue of a plane theorem, and the second theorem is its dual, that is, the result of interchanging great circles and their poles. [32] Kiper et al. [ 33 ] proved a converse of the theorem: If the summations of the opposite sides are equal in a spherical quadrilateral, then there exists an ...
Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof [ 1 ] : 411 of Feuerbach's theorem uses the converse theorem.
Proclus (410–485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four; in particular, he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However, he did give a postulate which is equivalent to the fifth postulate.
List of mathematical proofs; List of misnamed theorems; ... Euler's theorem in geometry ... Ptolemy's theorem ...
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.