A227879 -id:A227879 - cp's OEIS frontend

30, 50, 75, 120, 195, 200, 260, 270, 300, 340, 450, 480, 510, 525, 585, 675, 700, 750, 765, 780, 800, 845, 1014, 1015, 1040, 1080, 1156, 1200, 1250, 1360, 1365, 1470, 1530, 1554, 1740, 1755, 1800, 1845, 1875, 1920, 2040, 2100, 2210, 2220, 2275, 2340, 2430

Offset: 1

Michel Lagneau, Oct 26 2013

nonn

The excentral triangle, also called the tritangent triangle, of a triangle ABC is the triangle IJK with vertices corresponding to the excenters of ABC.
The excentral triangle has side lengths:
  a' = a*csc(A/2) where csc(z)=1/sin(z);
  b' = b*csc(B/2);
  c' = c*csc(C/2);
and area:
  A' = 4*A*a*b*c/((a+b-c)*(a-b+c)*(-a+b+c)).
Property of this sequence:
The areas of the original triangles are integers. The primitive triangles with areas a(n) are 30, 50, 75, 195, ...
The non-primitive triangles with areas 4*a(n) are in the sequence.
The following table gives the first values (A', A, a, b, c) where A' is the area of the excentral triangles, A is the area of the reference triangles ABC, a, b, c the integer sides of the original triangles ABC.
   ----------------------
   | A'|  A |  a|  b|  c|
   ----------------------
   | 30|   6|  3|  4|  5|
   | 50|  12|  5|  5|  6|
   | 75|  12|  5|  5|  8|
   |120|  24|  6|  8| 10|
   |195|  30|  5| 12| 13|
   |200|  48| 10| 10| 12|
   |260|  24|  4| 13| 15|
   |270|  54|  9| 12| 15|
   |300|  48| 10| 10| 16|
   |340|  60|  8| 15| 17|
   ......................

			30 is in the sequence because the area A' = 4*A*a*b*c/((a+b-c)*(a-b+c)*(-a+b+c)) of the excentral triangle corresponding to the initial triangle (3,4,5) is A' = 4*6*3*4*5/((3+4-5)*(3-4+5)*(-3+4+5)) = 30, where A = 6 obtained by Heron's formula A = sqrt(s*(s-a)*(s-b)*(s-c)) = sqrt(6*(6-3)*(6-4)*(6-5)) = 6, and where s = 6 is the semiperimeter.
The sides of the excentral triangle are:
  a' = 3*csc(1/2*arcsin(3/5)) = 9.48683298...
  b' = 4*csc(1/2*arcsin(4/5)) = 8.94427191...
  c' = 5*sqrt(2) = 7.07106781...

Clark Kimberling, Triangle Centers and Central Triangles. Congr. Numer. 129, 1-295, 1998.

Eric Weisstein's World of Mathematics, Excentral Triangles

Cf. A188158, A227879.

nn = 500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 && IntegerQ[4*Sqrt[area2]*a*b*c/((a + b - c)*(a - b + c)*(-a + b + c))], AppendTo[lst, 4*Sqrt[area2]*a*b*c/((a + b - c)*(a - b + c)*(-a + b + c))]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

cp's OEIS Frontend

A228097 Integer areas of excentral triangles of integer-sided triangles.

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