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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.

%I A242778 #25 Feb 16 2025 08:33:22
%S A242778 6,8,10,24,12,16,14,48,16,30,18,24,18,80,20,48,22,120,24,32,24,70,26,
%T A242778 168,28,96,30,40,30,72,30,224,32,60,32,126,34,288,36,48,36,160,38,360,
%U A242778 40,42,40,96,40,198,42,56,42,144,42,440,44,240,46,528,48,64
%N 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.
%C A242778 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.
%C A242778 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.
%C A242778 The circumradius R (the radius of the circumcircle) of any cyclic quadrilateral is given by
%C A242778 R = sqrt((ab+cd)(ac+bd)(ad+bc))/(4A).
%C A242778 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.
%C A242778 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.
%H A242778 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>
%e A242778 (a(1),a(2)) = (6,8) because, for (a,b,c,d) = (6,6,8,8) we obtain:
%e A242778 s = a + c = 6+8 = 14;
%e A242778 A = a*c = 6*8 = 48;
%e A242778 R = sqrt(a^2 + c^2)/2 = sqrt(6^2 + 8^2)/2 = 5.
%t A242778 nn=1500;lst={};Do[s=(2*a+2*c)/2;If[IntegerQ[s],area2=(s-a)^2*(s-c)^2;If[0<area2&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[(a^2+c^2)*4*a^2*c^2/((s-a)^2*(s-c)^2)]/4],AppendTo[lst, Union [{a},{c}]]]],{a,nn},{c,a}];Union[lst]
%Y A242778 Cf. A210250, A297790, A298860, A298907.
%K A242778 nonn
%O A242778 1,1
%A A242778 _Michel Lagneau_, May 22 2014
%E A242778 Definition and comments extended and/or corrected by _Gregory Gerard Wojnar_, Nov 10 2018