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 A219225 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Subsequence of A210250. 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)). REFERENCES Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. LINKS Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32. E. Gürel, Solution to Problem 1472, Maximal Area of Quadrilaterals, Math. Mag. 69 (1996), 149. Eric Weisstein's World of Mathematics, Cyclic Quadrilateral EXAMPLE 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. MATHEMATICA 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 *) CROSSREFS Cf. A210250. Sequence in context: A252072 A200856 A116301 * A045082 A257414 A179668 Adjacent sequences: A219222 A219223 A219224 * A219226 A219227 A219228 KEYWORD nonn AUTHOR Michel Lagneau, Nov 15 2012 EXTENSIONS Incorrect Mathematica program removed by Albert Lau, May 25 2016 Missing terms 18144, 20064, 21504 and more term from Albert Lau, May 25 2016 STATUS approved

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Last modified February 8 06:05 EST 2023. Contains 360134 sequences. (Running on oeis4.)