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A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common. [17]: p. 84
Robbins pentagons were named by Buchholz & MacDougall (2008) after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a function of its edge lengths. Buchholz and MacDougall chose this name by analogy with the naming of Heron triangles after Hero of Alexandria , the discoverer of Heron's formula for the ...
Every regular polygon is a cyclic polygon. For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, … are equal, and sides 2, 4, 6, … are equal). [11] A ...
More generally, an n-sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclic quadrilateral. All rectangles , isosceles trapezoids , right kites , and regular polygons are cyclic, but not every polygon is.
A cyclic polygon (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. [35]
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.
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