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

%I A228097 #29 Feb 24 2025 04:35:02
%S A228097 30,50,75,120,195,200,260,270,300,340,450,480,510,525,585,675,700,750,
%T A228097 765,780,800,845,1014,1015,1040,1080,1156,1200,1250,1360,1365,1470,
%U A228097 1530,1554,1740,1755,1800,1845,1875,1920,2040,2100,2210,2220,2275,2340,2430
%N A228097 Integer areas of excentral triangles of integer-sided triangles.
%C A228097 The excentral triangle, also called the tritangent triangle, of a triangle ABC is the triangle IJK with vertices corresponding to the excenters of ABC.
%C A228097 The excentral triangle has side lengths:
%C A228097   a' = a*csc(A/2) where csc(z)=1/sin(z);
%C A228097   b' = b*csc(B/2);
%C A228097   c' = c*csc(C/2);
%C A228097 and area:
%C A228097   A' = 4*A*a*b*c/((a+b-c)*(a-b+c)*(-a+b+c)).
%C A228097 Property of this sequence:
%C A228097 The areas of the original triangles are integers. The primitive triangles with areas a(n) are 30, 50, 75, 195, ...
%C A228097 The non-primitive triangles with areas 4*a(n) are in the sequence.
%C A228097 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.
%C A228097    ----------------------
%C A228097    | A'|  A |  a|  b|  c|
%C A228097    ----------------------
%C A228097    | 30|   6|  3|  4|  5|
%C A228097    | 50|  12|  5|  5|  6|
%C A228097    | 75|  12|  5|  5|  8|
%C A228097    |120|  24|  6|  8| 10|
%C A228097    |195|  30|  5| 12| 13|
%C A228097    |200|  48| 10| 10| 12|
%C A228097    |260|  24|  4| 13| 15|
%C A228097    |270|  54|  9| 12| 15|
%C A228097    |300|  48| 10| 10| 16|
%C A228097    |340|  60|  8| 15| 17|
%C A228097    ......................
%D A228097 Clark Kimberling, Triangle Centers and Central Triangles. Congr. Numer. 129, 1-295, 1998.
%H A228097 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExcentralTriangle.html">Excentral Triangles</a>
%e A228097 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.
%e A228097 The sides of the excentral triangle are:
%e A228097   a' = 3*csc(1/2*arcsin(3/5)) = 9.48683298...
%e A228097   b' = 4*csc(1/2*arcsin(4/5)) = 8.94427191...
%e A228097   c' = 5*sqrt(2) = 7.07106781...
%t A228097 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]
%Y A228097 Cf. A188158, A227879.
%K A228097 nonn
%O A228097 1,1
%A A228097 _Michel Lagneau_, Oct 26 2013