cp's OEIS Frontend

In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribing circle, and the vertices are said to be concyclic.

The area A of any cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula: A = sqrt((s - a)(s - b)(s - c)(s - d)) where s, the semiperimeter, is s= (a+b+c+d)/2.

The circumradius R (the radius of the circumcircle) of any cyclic quadrilateral is given by

R = sqrt((ab+cd)(ac+bd)(ad+bc))/(4A).

Many cyclic quadrilaterals [a, b, c, d] with integer sidelengths, integer area, and integer circumradius have the property that a = b and c = d, thus forming a kite with two right angles, with the long diagonal of the kite being a diameter of the circle; thus the circumradius is R = sqrt(a^2 + c^2)/2. Since Brahmagupta's formula is invariant upon permutation of the sides, the area of such a kite is the same as that of the rectangle with sides [a, c, b, d]. So in this case s = a+c, and A = a*c. In particular, the double of any Pythagorean triple will satisfy our requirements.

Nevertheless, there also exist cyclic quadrilaterals with integer sidelengths, integer area, and integer circumradius, whose four sides are distinct; for example, [a, b, c, d] = [ 14, 30, 40, 48] => A = 936 and R = 25.

Entries are listed as quadruples: (a,b,c,d). They are ordered first by perimeter, second by area, then by a, then b, then c, then d. Rectangles and kites with two right angles are not listed; thus a < b <= c <= d. By "primitive" we mean (a,b,c,d) is not a multiple of any earlier quadruple.

It appears that the number of odd sidelengths in any quadruple is always 0, 2, or 4.