A242778 Sides (a,c) of cyclic quadrilaterals of integer sides (a,b,c,d), integer areas, and integer circumradius such that a=b and c=d.

6, 8, 10, 24, 12, 16, 14, 48, 16, 30, 18, 24, 18, 80, 20, 48, 22, 120, 24, 32, 24, 70, 26, 168, 28, 96, 30, 40, 30, 72, 30, 224, 32, 60, 32, 126, 34, 288, 36, 48, 36, 160, 38, 360, 40, 42, 40, 96, 40, 198, 42, 56, 42, 144, 42, 440, 44, 240, 46, 528, 48, 64

Offset: 1

Author

Michel Lagneau, May 22 2014

Keywords

• nonn

Comments

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.

Examples

			(a(1),a(2)) = (6,8) because, for (a,b,c,d) = (6,6,8,8) we obtain:
s = a + c = 6+8 = 14;
A = a*c = 6*8 = 48;
R = sqrt(a^2 + c^2)/2 = sqrt(6^2 + 8^2)/2 = 5.
		

Links

• Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

Crossrefs

Cf. A210250, A297790, A298860, A298907.

Programs

Mathematica

nn=1500;lst={};Do[s=(2*a+2*c)/2;If[IntegerQ[s],area2=(s-a)^2*(s-c)^2;If[0
				

Extensions

Definition and comments extended and/or corrected by Gregory Gerard Wojnar, Nov 10 2018