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A242778
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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.
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4
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6, 8, 10, 24, 12, 16, 14, 48, 16, 30, 18, 24, 18, 80, 20, 48, 22, 120, 24, 32, 24, 70, 26, 168, 28, 96, 30, 40, 30, 72, 30, 224, 32, 60, 32, 126, 34, 288, 36, 48, 36, 160, 38, 360, 40, 42, 40, 96, 40, 198, 42, 56, 42, 144, 42, 440, 44, 240, 46, 528, 48, 64
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OFFSET
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1,1
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COMMENTS
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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.
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.
The circumradius R (the radius of the circumcircle) of any cyclic quadrilateral is given by
R = sqrt((ab+cd)(ac+bd)(ad+bc))/(4A).
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.
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.
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LINKS
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EXAMPLE
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(a(1),a(2)) = (6,8) because, for (a,b,c,d) = (6,6,8,8) we obtain:
s = a + c = 6+8 = 14;
A = a*c = 6*8 = 48;
R = sqrt(a^2 + c^2)/2 = sqrt(6^2 + 8^2)/2 = 5.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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