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A273771 Integer area of primitive tangential quadrilateral with integer sides, integer inradius and rational diagonals.

%I A273771 #12 Jun 17 2016 00:41:48
%S A273771 168,408,420,528,588,600,660,1008
%N A273771 Integer area of primitive tangential quadrilateral with integer sides, integer inradius and rational diagonals.
%C A273771 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 A273771 The area S of a tangential quadrilateral is given by S = r s where s is the semiperimeter and r is the inradius.
%C A273771 The sides of a tangential quadrilateral satisfy s = a + c = b + d where a,c and b,d are opposite sides.
%C A273771 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 A273771 As a consequences of above formula, a b c d >= S^2.
%C A273771 The diagonal separating ad and bc is p=Sqrt[(a-d)^2+(4S^2)/(a d+b c+2D)]
%C A273771 The diagonal separating ab and cd is q=Sqrt[(a-b)^2+(4S^2)/(a b+c d-2D)]
%e A273771 a,   b,   c,  d,  S,    r,  p,   q
%e A273771 15,  15,  13, 13, 168,  6,  14,  24
%e A273771 26,  26,  25, 25, 408,  8,  17,  48
%e A273771 25,  25,  17, 17, 420,  10, 28,  30
%e A273771 26,  26,  22, 22, 528,  11, 40,  132/5
%e A273771 28,  28,  21, 21, 588,  12, 35,  168/5
%e A273771 25,  25,  25, 25, 600,  12, 40,  30
%e A273771 39,  30,  16, 25, 660,  12, 34,  39
%e A273771 102, 102, 10, 10, 1008, 9,  104, 252/13
%t A273771 SMax=500;
%t A273771 Do[
%t A273771   If[a==c&&\[CapitalDelta]<0,Continue[]];
%t A273771   If[GCD[a,b,s,r]>1,Continue[]];
%t A273771   If[b c+\[CapitalDelta]<=0||c d-\[CapitalDelta]<=0,Continue[]];
%t A273771   If[!{p=Sqrt[(a-d)^2+(4S^2)/(a d+2\[CapitalDelta]+b c)],
%t A273771        q=Sqrt[(a-b)^2+(4S^2)/(a b-2\[CapitalDelta]+c d)]
%t A273771       }\[Element]Rationals,Continue[]];
%t A273771   S(*{a,b,c,d,S,r,\[CapitalDelta],p,q}*)//Sow;
%t A273771   ,{S,SMax},{s,S//Divisors//Select[#,#^2>=4S&]&},{r,{S/s}}
%t A273771   ,{a,s/2//Ceiling,s},{c,{s-a}}
%t A273771   ,{b,s/2//Ceiling,a},{d,{s-b}}
%t A273771   ,{\[CapitalDelta],Select[{1,-1}Sqrt[a b c d-S^2],IntegerQ]//Union}
%t A273771 ]//Reap//Last//Last(*//TableForm*)
%t A273771 {p,q}=.;
%K A273771 nonn
%O A273771 1,1
%A A273771 _Albert Lau_, Jun 03 2016