cp's OEIS Frontend

A231276 Integer areas of the inner vecten triangles of integer-sided triangles.

%I A231276 #19 Feb 16 2025 08:33:20
%S A231276 5,20,21,23,29,39,41,45,59,63,80,83,84,92,116,125,131,156,164,173,180,
%T A231276 189,203,207,227,236,237,245,252,257,261,269,320,329,332,336,351,368,
%U A231276 369,371,405,464,479,497,500,524,525,531,567,575,605,623,624,656,663
%N A231276 Integer areas of the inner vecten triangles of integer-sided triangles.
%C A231276 Consider the internal erection of three squares on the sides of a triangle ABC. These centers form a triangle IJK.
%C A231276 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.
%C A231276 Its side lengths are
%C A231276   a' = sqrt((b^2 + c^2 - 4*A)/2),
%C A231276   b' = sqrt((a^2 + c^2 - 4*A)/2),
%C A231276   c' = sqrt((a^2 + b^2 - 4*A)/2).
%C A231276 The circumcircle of the inner vecten circle is the inner vecten circle.
%C A231276 Properties of this sequence:
%C A231276 The primitive triangles are 5, 21, 23, 29, 39, 41, ...
%C A231276 The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
%C A231276 It appears that if the triangles are isosceles, one of the sides of the inner vecten triangles is an integer (see the table below).
%C A231276 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.
%C A231276 -----------------------------------------------------------------------
%C A231276 |  A' |  A   |  a  |   b  |  c   |     a'      |    b'       |  c'
%C A231276 -----------------------------------------------------------------------
%C A231276 |  5  |   48 |  10 |   10 |  12  |   sqrt(26)  |    sqrt(26) |   2
%C A231276 | 20  |  192 |  20 |   20 |  24  | 2*sqrt(26)  |  2*sqrt(26) |   4
%C A231276 | 21  |  240 |  20 |   20 |  26  |    14       |   sqrt(58)  | sqrt(58)
%C A231276 | 23  | 1680 |  48 |   74 |  74  |    46       |   sqrt(530) | sqrt(530)
%C A231276 | 29  | 1680 |  50 |   68 |  78  |   sqrt(1994)|  2*sqrt(233)| sqrt(202)
%C A231276 | 39  | 1680 |  58 |   58 |  80  |   sqrt(1522)|   sqrt(1522)|   2
%C A231276 | 41  |  336 |  26 |   28 |  30  |   sqrt(170) |  2*sqrt(29) | sqrt(58)
%C A231276 | 45  |  432 |  30 |   30 |  36  | 3*sqrt(26)  |  3*sqrt(26) |   6
%C A231276 | 59  | 1440 |  50 |   58 |  72  |   sqrt(1394)|    sqrt(962)| 2*sqrt(13)
%C A231276 | 63  |  480 |  32 |   34 |  34  |    14       |    sqrt(130)|  sqrt(130)
%C A231276 | 80  |  768 |  40 |   40 |  48  | 4*sqrt(26)  |  4*sqrt(26) |   8
%C A231276 | 83  | 2880 |  74 |   78 | 104  |   sqrt(2690)|   sqrt(2386)| 2*sqrt(5)
%C A231276 .............................................................
%D A231276 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.
%H A231276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InnerVectenTriangle.html">Inner Vecten Triangle</a>
%e A231276 5 is in the sequence. We use two ways:
%e A231276 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.
%e A231276 Second way: by calculation of the sides a', b', c' and by use of Heron's formula.
%e A231276   a’ = sqrt((b^2 + c^2 - 4*A)/2) = sqrt((10^2 + 12^2 - 4*48)/2) = sqrt(26);
%e A231276   b’ = sqrt((a^2 + c^2 - 4*A)/2) = sqrt((10^2 + 12^2 - 4*48)/2) = sqrt(26);
%e A231276   c’ = sqrt((a^2 + b^2 - 4*A)/2) = sqrt((10^2 + 10^2 - 4*48)/2) = 2.
%e A231276 Now we use Heron's formula with (a',b',c').
%e A231276 We find A' = sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:
%e A231276 s1 = (a' + b' + c')/2 = (sqrt(26) + sqrt(26) + 2)/2.
%e A231276 We find A' = 5.
%t A231276 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]
%Y A231276 Cf. A188158, A231275.
%K A231276 nonn
%O A231276 1,1
%A A231276 _Michel Lagneau_, Nov 06 2013