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A219225
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Area A of the cyclic quadrilaterals PQRS with PQ>=QR>=RS>=SP, such that A, the sides, the radius of the circumcircle and the two diagonals are integers.
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4
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768, 936, 1200, 2856, 3072, 3744, 4536, 4800, 5016, 5376, 6696, 6912, 7056, 7560, 7752, 8184, 8424, 9240, 10800, 11424, 11544, 12288, 12480, 12936, 14976, 16848, 18144, 18696, 19200, 19200, 20064, 21504, 23040, 23400, 24024, 25080, 25704, 25944, 26784, 27048, 27648, 27648, 27648, 27864, 28224, 28560, 30000, 30240, 31008, 32736, 33696, 34560, 36960, 36960, 37632, 40392, 40560, 40824, 41064, 41184, 42240, 42840, 43200
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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 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 diagonals of a cyclic quadrilateral have length:
p = sqrt((ab+cd)(ac+bd)/(ad+bc))
q = sqrt((ac+bd)(ad+bc)/(ab+cd)).
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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.
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LINKS
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EXAMPLE
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936 is in the sequence because, with sides (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;
p = sqrt((14*30+40*48)( 14*40+30*48)/( 14*48+30*40)) = 50;
q= sqrt((14*40+30*48)( 14*48+30*40)/( 14*30+40*48)) = 40.
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MATHEMATICA
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SMax=10000;
Do[
Do[
x=S^2/(u v w);
If[u+v+w+x//OddQ, Continue[]];
If[v+w+x<=u, Continue[]];
r=Sqrt[v w+u x]Sqrt[u w+v x]Sqrt[u v+w x]/(4S);
If[r//IntegerQ//Not, Continue[]];
{a, b, c, d}=(u+v+w+x)/2-{u, v, w, x};
If[4S r/(a b+c d)//IntegerQ//Not, Continue[]];
If[4S r/(a d+b c)//IntegerQ//Not, Continue[]];
(*{a, b, c, d, r, S}//Sow*);
S//Sow; Break[]; (*to generate a table, comment out this line and uncomment previous line*)
, {u, S^2//Divisors//Select[#, S<=#^2&]&}
, {v, S^2/u//Divisors//Select[#, S^2<=u#^3&&#<=u&]&}
, {w, S^2/(u v)//Divisors//Select[#, S^2<=u v#^2&&#<=v&]&}
]
, {S, 24, SMax, 24}
]//Reap//Last//Last
{x, r, a, b, c, d}=.; (* Albert Lau, May 25 2016 *)
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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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Incorrect Mathematica program removed by Albert Lau, May 25 2016
Missing terms 18144, 20064, 21504 and more term from Albert Lau, May 25 2016
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STATUS
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approved
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