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A228097
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Integer areas of excentral triangles of integer-sided triangles.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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| A'| A | a| b| c|
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| 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|
......................
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REFERENCES
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C. Kimberling, Triangle Centers and Central Triangles. Congr. Numer. 129, 1-295, 1998.
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LINKS
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EXAMPLE
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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...
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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