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A233315
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Integer areas A of integer-sided cyclic quadrilaterals such that the length of the circumradius is a perfect square.
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4
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672, 768, 936, 1200, 10752, 12288, 14976, 19200, 34560, 40560, 48840, 54432, 57120, 62208, 75816, 97200, 138720, 154560, 172032, 196608, 239616, 307200, 420000, 480000, 552960, 585000, 648960, 750000, 781440, 870912, 913920, 995328, 1213056, 1555200, 2219520
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OFFSET
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1,1
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COMMENTS
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Subset of A210250. The corresponding square circumradius are 25,25,25,25,100,100,100,100,169,169,169,169,225, ...
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.
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.
The circumradius R (the radius of the circumcircle) is given by R = sqrt((ab+cd)(ac+bd)(ad+bc))/4A.
The areas A of the primitive quadrilaterals of sides (a,b,c,d) are 672,768,936,1200,34560,40560,48840,57120,...
The areas of the non-primitive quadrilaterals of sides (a*p^2, b*p^2, c*p^2, d*p^2) are in the sequence with the value A*p^4.
The following table gives the first values (A, a, b, c, d, R) where A is the integer area, R the radius of the circumcircle, and a, b, c, d are the integer sides of the cyclic quadrilateral.
+-------+-----+-----+-----+-----+-----+
| A | a | b | c | d | R |
+-------+-----+-----+-----+-----+-----+
| 672 | 14 | 14 | 48 | 48 | 25 |
| 768 | 14 | 30 | 30 | 50 | 25 |
| 936 | 14 | 30 | 40 | 48 | 25 |
| 1200 | 30 | 30 | 40 | 40 | 25 |
| 10752 | 56 | 56 | 192 | 192 | 100 |
| 12288 | 56 | 120 | 120 | 200 | 100 |
| 14976 | 56 | 120 | 160 | 192 | 100 |
| 19200 | 120 | 120 | 160 | 160 | 100 |
| 34560 | 130 | 130 | 238 | 338 | 169 |
| 40560 | 130 | 130 | 312 | 312 | 169 |
| 48840 | 130 | 238 | 240 | 312 | 169 |
| 54432 | 126 | 126 | 432 | 432 | 225 |
| 57120 | 238 | 238 | 240 | 240 | 169 |
| 62208 | 126 | 270 | 270 | 450 | 225 |
| 75816 | 126 | 270 | 360 | 432 | 225 |
| 97200 | 270 | 270 | 360 | 360 | 225 |
.......................................
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REFERENCES
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Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
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LINKS
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EXAMPLE
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936 is in the sequence because, for (a,b,c,d) = (14,30,40,48) we obtain:
s = (14+30+40+48)/2 = 66;
A = sqrt((66-14)*(66-30)*(66-40)*(66-48))=936;
R = sqrt((14*30+40*48)*(14*40+30*48)*(14*48+30*40))/(4*936) = 93600/3744 = 25 is square.
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MATHEMATICA
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nn=500; 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[Sqrt[(a*b+c*d)*(a*c+b*d)*(a*d+b*c)/area2]/4]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}, {d, c}]; Union[lst]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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