%I #12 Jun 17 2016 00:41:48
%S 168,408,420,528,588,600,660,1008
%N Integer area of primitive tangential quadrilateral with integer sides, integer inradius and rational diagonals.
%C A tangential quadrilateral is a quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius.
%C The area S of a tangential quadrilateral is given by S = r s where s is the semiperimeter and r is the inradius.
%C The sides of a tangential quadrilateral satisfy s = a + c = b + d where a,c and b,d are opposite sides.
%C Let D^2 = a b c d - S^2 (D can be positive or negative), then the distance from the tangent point on a(or b) to the vertex point between a,b is given by (ab-D)/s. Similar formula is given for changing a-b to b-c, c-d and d-a.
%C As a consequences of above formula, a b c d >= S^2.
%C The diagonal separating ad and bc is p=Sqrt[(a-d)^2+(4S^2)/(a d+b c+2D)]
%C The diagonal separating ab and cd is q=Sqrt[(a-b)^2+(4S^2)/(a b+c d-2D)]
%e a, b, c, d, S, r, p, q
%e 15, 15, 13, 13, 168, 6, 14, 24
%e 26, 26, 25, 25, 408, 8, 17, 48
%e 25, 25, 17, 17, 420, 10, 28, 30
%e 26, 26, 22, 22, 528, 11, 40, 132/5
%e 28, 28, 21, 21, 588, 12, 35, 168/5
%e 25, 25, 25, 25, 600, 12, 40, 30
%e 39, 30, 16, 25, 660, 12, 34, 39
%e 102, 102, 10, 10, 1008, 9, 104, 252/13
%t SMax=500;
%t Do[
%t If[a==c&&\[CapitalDelta]<0,Continue[]];
%t If[GCD[a,b,s,r]>1,Continue[]];
%t If[b c+\[CapitalDelta]<=0||c d-\[CapitalDelta]<=0,Continue[]];
%t If[!{p=Sqrt[(a-d)^2+(4S^2)/(a d+2\[CapitalDelta]+b c)],
%t q=Sqrt[(a-b)^2+(4S^2)/(a b-2\[CapitalDelta]+c d)]
%t }\[Element]Rationals,Continue[]];
%t S(*{a,b,c,d,S,r,\[CapitalDelta],p,q}*)//Sow;
%t ,{S,SMax},{s,S//Divisors//Select[#,#^2>=4S&]&},{r,{S/s}}
%t ,{a,s/2//Ceiling,s},{c,{s-a}}
%t ,{b,s/2//Ceiling,a},{d,{s-b}}
%t ,{\[CapitalDelta],Select[{1,-1}Sqrt[a b c d-S^2],IntegerQ]//Union}
%t ]//Reap//Last//Last(*//TableForm*)
%t {p,q}=.;
%K nonn
%O 1,1
%A _Albert Lau_, Jun 03 2016