login
A231276
Integer areas of the inner vecten triangles of integer-sided triangles.
1
5, 20, 21, 23, 29, 39, 41, 45, 59, 63, 80, 83, 84, 92, 116, 125, 131, 156, 164, 173, 180, 189, 203, 207, 227, 236, 237, 245, 252, 257, 261, 269, 320, 329, 332, 336, 351, 368, 369, 371, 405, 464, 479, 497, 500, 524, 525, 531, 567, 575, 605, 623, 624, 656, 663
OFFSET
1,1
COMMENTS
Consider the internal erection of three squares on the sides of a triangle ABC. These centers form a triangle IJK. The area of the inner vecten triangle is
A' = A - (a^2 + b^2 + c^2)/8,
where A is the area of the reference triangle. Its side lengths are
a' = sqrt((b^2 + c^2 - 4*A)/2),
b' = sqrt((a^2 + c^2 - 4*A)/2),
c' = sqrt((a^2 + b^2 - 4*A)/2).
The circumcircle of the inner vecten circle is the inner vecten circle.
Properties of this sequence:
The primitive triangles are 5, 21, 23, 29, 39, 41, ...
The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
It appears that if the triangles are isosceles, one of the sides of the inner vecten triangles is integer (see the table below).
The following table gives the first values (A, A', a, b, c, a', b', c') where A is the area of the initial triangles, A' is the area of the inner vecten triangles, a, b, c are the integer sides of the initial triangles, and a', b', c' are the sides of the inner vecten triangles.
-----------------------------------------------------------------------
| A' | A | a | b | c | a' | b' | c'
-----------------------------------------------------------------------
| 5 | 48 | 10 | 10 | 12 | sqrt(26) | sqrt(26) | 2
| 20 | 192 | 20 | 20 | 24 | 2*sqrt(26) | 2*sqrt(26) | 4
| 21 | 240 | 20 | 20 | 26 | 14 | sqrt(58) | sqrt(58)
| 23 | 1680 | 48 | 74 | 74 | 46 | sqrt(530) | sqrt(530)
| 29 | 1680 | 50 | 68 | 78 | sqrt(1994)| 2*sqrt(233)| sqrt(202)
| 39 | 1680 | 58 | 58 | 80 | sqrt(1522)| sqrt(1522)| 2
| 41 | 336 | 26 | 28 | 30 | sqrt(170) | 2*sqrt(29) | sqrt(58)
| 45 | 432 | 30 | 30 | 36 | 3*sqrt(26) | 3*sqrt(26) | 6
| 59 | 1440 | 50 | 58 | 72 | sqrt(1394)| sqrt(962)| 2*sqrt(13)
| 63 | 480 | 32 | 34 | 34 | 14 | sqrt(130)| sqrt(130)
| 80 | 768 | 40 | 40 | 48 | 4*sqrt(26) | 4*sqrt(26) | 8
| 83 | 2880 | 74 | 78 | 104 | sqrt(2690)| sqrt(2386)| 2*sqrt(5)
.............................................................
REFERENCES
H. S. M. Coxeter and S. L. Greitzer, Points and Lines Connected with a Triangle, Ch. 1 in Geometry Revisited, Washington DC, Math. Assoc. Amer., pp. 1-26 and 96-97, 1967.
EXAMPLE
5 is in the sequence. We use two ways:
First way: with the triangle (10, 10, 12) the formula A' = A - (a^2 + b^2 + c^2)/8 gives directly the result: A' = 48 - (10^2 + 10^2 + 12^2)/8 = 5 where the area A = 48 is obtained by Heron's formula A = sqrt(s*(s-a)*(s-b)*(s-c)) = sqrt(16*(16-10)*(16-10)*(16-12)) = 48, where s is the semiperimeter.
Second way: by calculation of the sides a', b', c' and by use of Heron's formula.
a’ = sqrt((b^2 + c^2 - 4*A)/2) = sqrt((10^2 + 12^2 - 4*48)/2) = sqrt(26);
b’ = sqrt((a^2 + c^2 - 4*A)/2) = sqrt((10^2 + 12^2 - 4*48)/2) = sqrt(26);
c’ = sqrt((a^2 + b^2 - 4*A)/2) = sqrt((10^2 + 10^2 - 4*48)/2) = 2.
Now we use Heron's formula with (a',b',c'). We find A' = sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:
s1 = (a' + b' + c')/2 = (sqrt(26) + sqrt(26) + 2)/2.
We find A' = 5.
MATHEMATICA
nn = 500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); t = (a^2 + b^2 + c^2)/8; If[0 < area2 && Sqrt[area2] - t > 0 && IntegerQ[Sqrt[area2] - t], AppendTo[lst, Sqrt[area2] - t]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
CROSSREFS
Sequence in context: A243800 A335555 A098047 * A101728 A053240 A034123
KEYWORD
nonn
AUTHOR
Michel Lagneau, Nov 06 2013
STATUS
approved