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Moreover, the converse of Ptolemy's theorem is also true: In a quadrilateral, if the sum of the products of the lengths of its two pairs of opposite sides is equal to the product of the lengths of its diagonals, then the quadrilateral can be inscribed in a circle i.e. it is a cyclic quadrilateral .
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 ...
Proof of law of cosines using Ptolemy's theorem. Referring to the diagram, triangle ABC with sides AB = c, BC = a and AC = b is drawn inside its circumcircle as shown. Triangle ABD is constructed congruent to triangle ABC with AD = BC and BD = AC. Perpendiculars from D and C meet base AB at E and F respectively.
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 ...
Donald Trump said in an interview with Fox News that Ukrainian President Volodymyr Zelenskyy "shouldn't have allowed" war in his country to happen.
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 ]
Naomi Osaka's second-round match at the Australian Open could not possibly have started in a worse way. Osaka strode to the sideline for the changeover, plopped herself down and draped a white ...
Today's NYT Connections puzzle for Wednesday, January 15, 2025The New York Times