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

168, 408, 420, 528, 588, 600, 660, 1008

Offset: 1

Author

Albert Lau, Jun 03 2016

Keywords

• nonn

Comments

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.

The area S of a tangential quadrilateral is given by S = r s where s is the semiperimeter and r is the inradius.

The sides of a tangential quadrilateral satisfy s = a + c = b + d where a,c and b,d are opposite sides.

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.

As a consequences of above formula, a b c d >= S^2.

The diagonal separating ad and bc is p=Sqrt[(a-d)^2+(4S^2)/(a d+b c+2D)]

The diagonal separating ab and cd is q=Sqrt[(a-b)^2+(4S^2)/(a b+c d-2D)]

Examples

			a,   b,   c,  d,  S,    r,  p,   q
15,  15,  13, 13, 168,  6,  14,  24
26,  26,  25, 25, 408,  8,  17,  48
25,  25,  17, 17, 420,  10, 28,  30
26,  26,  22, 22, 528,  11, 40,  132/5
28,  28,  21, 21, 588,  12, 35,  168/5
25,  25,  25, 25, 600,  12, 40,  30
39,  30,  16, 25, 660,  12, 34,  39
102, 102, 10, 10, 1008, 9,  104, 252/13
		

Programs

Mathematica

SMax=500;
Do[
  If[a==c&&\[CapitalDelta]<0,Continue[]];
  If[GCD[a,b,s,r]>1,Continue[]];
  If[b c+\[CapitalDelta]<=0||c d-\[CapitalDelta]<=0,Continue[]];
  If[!{p=Sqrt[(a-d)^2+(4S^2)/(a d+2\[CapitalDelta]+b c)],
       q=Sqrt[(a-b)^2+(4S^2)/(a b-2\[CapitalDelta]+c d)]
      }\[Element]Rationals,Continue[]];
  S(*{a,b,c,d,S,r,\[CapitalDelta],p,q}*)//Sow;
  ,{S,SMax},{s,S//Divisors//Select[#,#^2>=4S&]&},{r,{S/s}}
  ,{a,s/2//Ceiling,s},{c,{s-a}}
  ,{b,s/2//Ceiling,a},{d,{s-b}}
  ,{\[CapitalDelta],Select[{1,-1}Sqrt[a b c d-S^2],IntegerQ]//Union}
]//Reap//Last//Last(*//TableForm*)
{p,q}=.;