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A233315
Integer areas A of integer-sided cyclic quadrilaterals such that the length of the circumradius is a perfect square.
4
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
OFFSET
1,1
COMMENTS
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 |
.......................................
REFERENCES
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.
EXAMPLE
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.
MATHEMATICA
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]
CROSSREFS
Cf. A210250.
Sequence in context: A281526 A053085 A057695 * A157363 A308574 A234732
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 07 2013
STATUS
approved