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A218431 Cyclic quadrilateral numbers: numbers m = a*b*c*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.

%I A218431 #29 Feb 16 2025 08:33:18
%S A218431 2304,36864,57600,186624,230400,451584,589824,630000,806400,921600,
%T A218431 1440000,2073600,2822400,2985984,3686400,4665600,5531904,6969600,
%U A218431 7225344,8960000,9437184,10080000,12672000,12902400,14745600,15116544,16257024,18662400,19079424
%N A218431 Cyclic quadrilateral numbers: numbers m = a*b*c*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.
%C A218431 A cyclic quadrilateral is a quadrilateral for which a circle can be circumscribed so that it touches each polygon vertex.
%C A218431 A cyclic quadrilateral number m is an integer with at least one decomposition m = a*b*c*d such that the area of the quadrilateral of sides (a,b,c,d) and the circumradius are integers. Because this property is not always unique, we introduce the notion of "cyclic quadrilateral order" for each cyclic quadrilateral number m, denoted by CQO(m). For example, CQO(2304) = 1 because the decomposition 2304 = 8*8*6*6 is unique with the quadrilateral (8,8,6,6) whose area A is given by Brahmagupta's formula: A = sqrt((s - a)*(s - b)*(s - c)*(s - d)) where the semiperimeter is s = (a+b+c+d)/2 and the circumradius R (the radius of the circumcircle) is given by R = sqrt((ab+cd)*(ac+bd)*(ad+bc))/(4A) => A = sqrt((14-8)*(14-8)*(14-6)*(14-6)) = 48, and R = 5, but CQO(2822400) = 2 because 2822400 = 24*24*70*70 = 40*40*42*42 and the area of the quadrilateral (24,24,70,70) equals 1680 with R = 37 and the area of the quadrilateral (40,40,42,42) also equals 1680 with R = 29.
%C A218431 The number of ways to write m = a*b*c*d with 1 <= a <= b <= c <= d <= m is given by A218320, thus: CQO(m) <= A218320(m).
%C A218431 If m is in this sequence, so is m*k^4 for any k > 0. Thus this sequence is infinite.
%C A218431 In view of the preceding comment, one might call "primitive" the terms m of the sequence for which there is no k > 1 such that m/k^4 is again a term of the sequence. These terms are 2304, 57600, 230400, 451584, 630000, ...
%H A218431 E. Gürel, <a href="http://www.jstor.org/stable/2690677?seq=7">Solution to Problem 1472, Maximal Area of Quadrilaterals</a>, Math. Mag. 69 (1996), 149.
%H A218431 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/CyclicQuadrilateral.html">MathWorld: Cyclic Quadrilateral</a>.
%e A218431 2304 is in the sequence because 2304 = 8*8*6*6 and we obtain:
%e A218431 s = (8+8+6+6)/2 = 14;
%e A218431 A = sqrt((14-8)*(14-8)*(14-6)*(14-6)) = 48;
%e A218431 R = sqrt((8*8 + 6*6)*(8*6 + 8*6)*(8*6 + 8*6))/(4*48) = 5.
%t A218431 nn=200; lst={}; Do[s=(a+b+c+d)/2; If[IntegerQ[s], area2=(s-a)*(s-b)*(s-c)*(s-d); If[0<area2&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[(a*b+c*d)*(a*c+b*d)*(a*d+b*c)/((s-a)*(s-b)*(s-c)*(s-d))]/4], AppendTo[lst, a*b*c*d]]], {a, nn}, {b, a}, {c, b}, {d, c}]; Union[lst]
%Y A218431 Cf. A210250.
%K A218431 nonn,nice
%O A218431 1,1
%A A218431 _Michel Lagneau_, Oct 28 2012
%E A218431 Typos in comment fixed by _Zak Seidov_ and _M. F. Hasler_, Sep 21 2013, Sep 21 2013