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Examples of cyclic quadrilaterals. In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral. = + 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 special case of the theorem for quadrilaterals states that the two pairs of opposite incircles of the theorem above have equal sums of radii. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral.
The quadrilateral case follows from a simple extension of the Japanese theorem for cyclic quadrilaterals, which shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry. [2]
In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. [1] It is named after the Indian mathematician Brahmagupta (598-668). [2]
In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
Ptolemy's inequality is often stated for a special case, in which the four points are the vertices of a convex quadrilateral, given in cyclic order. [2] [3] However, the theorem applies more generally to any four points; it is not required that the quadrilateral they form be convex, simple, or even planar.
The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
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