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%I #6 Jun 17 2016 00:43:12
%S 192,234,300,432,714,768,936,1134,1200,1254,1344,1674,1728,1764,1890,
%T 1938,2046,2106,2226,2310,2352,2700,2856,2886,3072,3120,3234,3744,
%U 3888,3990,4092,4212,4368,4536,4674,4800,4914,5016,5292,5376,5760,5850,6006,6270,6426
%N Integer area A of the cyclic quadrilaterals such that A, the sides and the two diagonals are integers.
%C The areas of the primitive cyclic quadrilaterals of this sequence are in A273691.
%C This sequence contains A233315 (768, 936, 1200,...).
%C In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed, and the vertices are said to be concyclic.
%C The area A of a 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 In a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, the lengths of the diagonals p = AC and q = BD can be expressed in terms of the sides as
%C p = sqrt((ac+bd)(ad+bc)/(ab+cd)) and q = sqrt((ac+bd)(ab+cd)/(ad+bc)).
%C The circumradius R (the radius of the circumcircle) is given by :
%C R = sqrt((ab+cd)(ac+bd)(ad+bc))/4A.
%C The corresponding sides of a(n) are not unique, for example for a(6) = 768 => (a,b,c,d) = (25, 25, 25, 39) or (a,b,c,d) = (14, 30, 30, 50).
%C The following table gives the first values (A, a, b, c, d, p, q, R) where A is the integer area, a, b, c, d are the integer sides of the cyclic quadrilateral, p, q are the integer diagonals, and R .
%C +--------+-------+-------+-------+--------+-------+------+-------+
%C | A | a | b | c | d | p | q | R |
%C +--------+-------+-------+-------+--------+-------+------+-------+
%C | 192 | 7 | 15 | 15 | 25 | 20 | 24 | 25/2 |
%C | 234 | 7 | 15 | 20 | 24 | 20 | 25 | 25/2 |
%C | 300 | 15 | 15 | 20 | 20 | 24 | 25 | 25/2 |
%C | 432 | 11 | 25 | 25 | 25 | 30 | 30 | 125/8 |
%C | 714 | 16 | 25 | 33 | 60 | 39 | 52 | 65/2 |
%C | 768 | 25 | 25 | 25 | 39 | 40 | 40 | 125/6 |
%C | 768 | 14 | 30 | 30 | 50 | 40 | 48 | 25 |
%C | 936 | 14 | 30 | 40 | 48 | 40 | 50 | 25 |
%C | 1134 | 16 | 25 | 52 | 65 | 39 | 63 | 65/2 |
%C | 1200 | 30 | 30 | 40 | 40 | 48 | 50 | 25 |
%C | 1254 | 16 | 25 | 60 | 63 | 39 | 65 | 65/2 |
%C | 1344 | 25 | 33 | 39 | 65 | 52 | 60 | 65/2 |
%C ..................................................................
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>
%e 192 is in the sequence because, for (a,b,c,d) = (7,15,15,25) we find:
%e s = (7+15+15+25)/2 = 31;
%e A = sqrt((31-7)(31-15)(31-15)(31-25)) = 192;
%e p = sqrt((7*15+15*25)*(7*25+15*15)/(7*15+15*25)) = 20;
%e q = sqrt((7*15+15*25)*(7*15+15*25)/(7*25+15*15)) = 24.
%t nn=200; lst={}; Do[s=(a+b+c+d)/2; If[IntegerQ[s], area2=(s-a)*(s-b)*(s-c)*(s-d); d1=Sqrt[(a*c+b*d)*(a*d+b*c)/(a*b+c*d)];d2=Sqrt[(a*c+b*d)*(a*b+c*d)/(a*d+b*c)];If[0<area2 && IntegerQ[Sqrt[area2]] && IntegerQ[d1]&& IntegerQ[d2], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}, {d, c}]; Union[lst]
%Y Cf. A210250, A218431, A219225, A230136, A233315, A242778, A273691.
%K nonn
%O 1,1
%A _Michel Lagneau_, Jun 02 2016
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