OFFSET
1,1
COMMENTS
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|
......................
REFERENCES
C. Kimberling, Triangle Centers and Central Triangles. Congr. Numer. 129, 1-295, 1998.
LINKS
Wolfram MathWorld, Excentral Triangles
EXAMPLE
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...
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Oct 26 2013
STATUS
approved