A210250 -id:A210250 - cp's OEIS frontend

A219192 Area A of the bicentric quadrilaterals such that A, the sides, the radius of the circumcircle and the radius of the incircle are integers.

2352, 9408, 21168, 37632, 58800, 69360, 84672, 115248, 150528, 190512, 235200, 253920, 277440, 284592, 338688, 397488, 460992, 529200, 602112, 624240, 645792, 679728, 762048, 849072, 940800, 1015680, 1037232, 1109760, 1138368, 1244208, 1354752, 1470000, 1589952

Offset: 1

Michel Lagneau, Nov 14 2012

nonn

Also numbers n such that there exists a decomposition n^2 = a*b*c*d where  a,b,c,d are the sides of a bicentric quadrilateral with the area, the inradius and the circumradius integers.
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. If the sides are a, b, c, d, then the area is given by A = sqrt(a*b*c*d). The inradius r of a bicentric quadrilateral is determined by the sides a, b, c, d according to r = sqrt(a*b*c*d)/(a+c) = sqrt(a*b*c*d)/(b+d). The circumradius R (the radius of the circumcircle) is given by R = sqrt((ab+cd)(ac+bd)(ad+bc))/4A.
If n is in this sequence, so is n*k^2 for any k > 0. Thus this sequence is infinite.
In view of the preceding comment, one might call "primitive" the elements of the sequence for which there is no k>1 such that n/k^2 is again a term of the sequence. These elements are 2352, 69360, 253920, 645792,... are listed in A219193.

			2352 is in the sequence because, with sides (a,b,c,d) = (56,56,42,42) we obtain :
s = (56+56+42+42)/2 = 98;
A = sqrt(56*56*42*42) = 2352 = sqrt((98-56)(98-56)(98-42)(98-42)) (Brahmagupta’s Formula);
r = 2352/(56+42) =24.
R = sqrt((56*56+42*42)(56*42+56*42)(56*42+56*42))/(4*2352) = 35.

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, 149, 1996.
Martin Josefsson, The area of a Bicentric Quadrilateral, Forum Geometricum (2011) 11:155-164.
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
Eric Weisstein's World of Mathematics, Bicentric Quadrilateral

Cf. A210250, A219193.

with(numtheory):nn:=15000:for a from 1 to nn do: b:=a: for c from b to nn do: for d from c to c while(sqrt(a*b*c*d)=floor(sqrt(a*b*c*d))) do:s:=(a+b+c+d)/2:a1:=(s-a)*(s-b)*(s-c)*(s-d):a2:=sqrt(a*b*c*d):r1:=a2/(a+c):r2:=a2/(b+d):rr:= sqrt((a*b+c*d) * (a*c+b*d) * (a*d+b*c))/(4*a2):if a1>0 and floor(sqrt(a1))=sqrt(a1) and a2 =floor(a2) and a2=sqrt(a1) and r1=floor(r1) and r2=floor(r2) and r1=r2 and rr =floor(rr) then printf ( "%d %d %d %d %d %d %d\n",a2,a,b,c,d,r1,rr):else fi:od:od:od:

nn=15000;lst={};Do[s=(2*a+2*d)/2;If[IntegerQ[s],area2=(s-a)*(s-a)*(s-d)*(s-d);area22=a*a*d*d;If[0

A219193 Primitive bicentric quadrilaterals defined in A219192.

Original entry on oeis.org

2352, 69360, 253920, 645792, 2824080, 3457440, 3711120, 13308240, 25163712, 26858160, 58552032, 112031040

Offset: 1

Michel Lagneau, Nov 14 2012

nonn

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The sequence A219192 gives the areas A of bicentric quadrilateral with sides (a,b,c,d), inradius r and circumradius R integer. For any such A, A*k^2 is again an element of A219192 for any integer k>0. Terms in A219192 which are not of that form (with k>1) are called "primitive", and listed here.
The term A219192(2) = 9408 is not in this sequence since 9408 = 2^2*2352 and 2352 = a(1).
The term n = 253920 = 2 ^ 5 * 3 * 5 * 23 ^ 2 is divisible by 4^2 and 23^2, but neither of n/3^2, n/5^2, n/23^2 is in A219192, therefore n is in this sequence.

Cf. A210250, A219192.

The Maple program gives the vector (A, a, b, c, d, r, R).
A is the area, a,b,c,d are the sides of the quadrilateral, r is the inradius and R is the circumradius.
with(numtheory):k:=1:T:=array(1..5000):T[k]:=2352:kk:=0:nn:=15000:for a from 1 to nn do: b:=a: for c from b to nn do: for d from c to c while(sqrt(a*b*c*d)=floor(sqrt(a*b*c*d))) do:s:=(a+b+c+d)/2:a1:=(s-a)*(s-b)*(s-c)*(s-d):a2:=sqrt(a*b*c*d):r1:=a2/(a+c):r2:=a2/(b+d):rr:= sqrt((a*b+c*d) * (a*c+b*d) * (a*d+b*c))/(4*a2):if a1>0 and floor(sqrt(a1))=sqrt(a1) and a2 =floor(a2) and a2=sqrt(a1) and r1=floor(r1) and r2=floor(r2) and r1=r2 and rr =floor(rr) then for j from 1 to k do: if sqrt(a2/T[j])=floor(sqrt(a2/T[j]) ) then kk:=1:else fi:od:if kk=0 then k:=k+1:T[k]:=a2: printf ( "%d %d %d %d %d %d %d\n",a2,a,b,c,d,r1,rr):else fi:kk:=0:fi:od:od:od:

cp's OEIS Frontend

A219192 Area A of the bicentric quadrilaterals such that A, the sides, the radius of the circumcircle and the radius of the incircle are integers.

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