A273752 - OEIS

%I #13 Aug 17 2018 19:47:27

%S 7140,16380,87780,1543668,1697892,4444440,5858580

%N Integer area of primitive bicentric quadrilateral with integer side, rational inradius and rational circumradius. Excluding right kites.

%C Bicentric quadrilaterals have the following properties:

%C 1. a+c = b+d = s where s is the semiperimeter;

%C 2. A+C = B+D = 180 degrees;

%C 2. Area S = sqrt(a b c d);

%C 3. Circumradius R = sqrt(a*b + c*d)*sqrt(a*c + b*d)*sqrt(a*d + b*c)/S;

%C 4. Inradius r = S/s (it follows that r is always rational if sides and area are integers);

%C 5. Length of the diagonal separating a-b and c-d is (4S*R)/(a*b + c*d), the other diagonal can be obtained by swapping b,c or swapping b,d. It follows that all diagonals are rational iff a,b,c,d,R,S are rationals.

%C There are only 7 primitive cases which are not right kites for S < 10^7.

%C From empirical observation, the area seems to be a multiple of 84. (If proven, the program could be modified to run 84 times as fast.)

%C Special cases of bicentric quadrilaterals are right kites and isosceles trapezium.

%C Integer right kites can be generated by joining two (a,b,c) Pythagorean triangles, which gives S=a b/2, R=c/2, r=ab/(a+b+c).

%C Integer isosceles trapezium is impossible. Proof:

%C 1. Let the sides of integer isosceles trapezium be (s-t,s,s+t,s);

%C 2. S = s*sqrt(s^2 - t^2) and R = 2*s^2*sqrt(2s^2 - t^2)/S;

%C 3. s^2 - t^2 and 2s^2 - t^2 are perfect squares;

%C 4. Let u^2 = 2s^2 - t^2, v^2 = s^2 - t^2;

%C 5. t^2,s^2,u^2 is an arithmetic progression with common difference = v^2;

%C 6. Fermat's right triangle theorem states that no integer solution exists, except v=0 which corresponds to (0,s,2s,s), a degenerate quadrilateral. QED.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bicentric_quadrilateral">Bicentric quadrilateral</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem">Fermat's right triangle theorem</a>.

%e All examples with S < 10^7:

%e a,    b,    c,    d,    S,       R,      r

%e 204,  140,  85,   21,   7140,    442,    476/15

%e 315,  260,  91,   36,   16380,   650,    140/3

%e 440,  399,  231,  190,  87780,   1885/2, 418/3

%e 2397, 1564, 1316, 483,  1543668, 4810,   128639/240

%e 4756, 3451, 1428, 123,  1697892, 15130,  348

%e 2849, 2184, 2145, 1480, 4444440, 6290,   3080/3

%e 5460, 5365, 1131, 1036, 5858580, 11050,  7215/8

%t SMin=7140;

%t SMax=16380(*WARNING: runs very slow*);

%t dS=1(*assuming S mod 84 = 0, set to 84 to run faster*);

%t Do[

%t   s=(a+b)/2+Sqrt[(a-b)^2/4+S^2/(a b)];

%t   If[s//IntegerQ//Not,Continue[]];

%t   If[GCD[a,b,s]>1,Continue[]];

%t   R=(Sqrt[#1#2+#3#4]Sqrt[#1#3+#2#4]Sqrt[#1#4+#2#3])/S&[a,b,s-b,s-a];

%t   If[R\[NotElement]Rationals,Continue[]];

%t   S(*{a,b,s-b,s-a,S,R,S/s}*)//Sow;

%t   ,{S,Round[SMin,dS],SMax,dS}

%t   ,{a,S^2//Divisors//Select[#,S<#^2&&#<S&]&}

%t   ,{b,S^2/a//Divisors//Select[#,a/2<#<a&&1+a-#<=S^2/(a#)<=#(2#-a)&]&}

%t ]//Reap//Last//Last(*//TableForm*)

%t {S,R,a,b,s}=.;

%K nonn,more

%O 1,1

%A _Albert Lau_, May 29 2016