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Integer areas of the inner vecten triangles of integer-sided triangles.
1

%I #18 Jan 10 2025 12:38:50

%S 5,20,21,23,29,39,41,45,59,63,80,83,84,92,116,125,131,156,164,173,180,

%T 189,203,207,227,236,237,245,252,257,261,269,320,329,332,336,351,368,

%U 369,371,405,464,479,497,500,524,525,531,567,575,605,623,624,656,663

%N Integer areas of the inner vecten triangles of integer-sided triangles.

%C Consider the internal erection of three squares on the sides of a triangle ABC. These centers form a triangle IJK.

%C 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 Its side lengths are

%C a' = sqrt((b^2 + c^2 - 4*A)/2),

%C b' = sqrt((a^2 + c^2 - 4*A)/2),

%C c' = sqrt((a^2 + b^2 - 4*A)/2).

%C The circumcircle of the inner vecten circle is the inner vecten circle.

%C Properties of this sequence:

%C The primitive triangles are 5, 21, 23, 29, 39, 41, ...

%C The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.

%C 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 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 -----------------------------------------------------------------------

%C | A' | A | a | b | c | a' | b' | c'

%C -----------------------------------------------------------------------

%C | 5 | 48 | 10 | 10 | 12 | sqrt(26) | sqrt(26) | 2

%C | 20 | 192 | 20 | 20 | 24 | 2*sqrt(26) | 2*sqrt(26) | 4

%C | 21 | 240 | 20 | 20 | 26 | 14 | sqrt(58) | sqrt(58)

%C | 23 | 1680 | 48 | 74 | 74 | 46 | sqrt(530) | sqrt(530)

%C | 29 | 1680 | 50 | 68 | 78 | sqrt(1994)| 2*sqrt(233)| sqrt(202)

%C | 39 | 1680 | 58 | 58 | 80 | sqrt(1522)| sqrt(1522)| 2

%C | 41 | 336 | 26 | 28 | 30 | sqrt(170) | 2*sqrt(29) | sqrt(58)

%C | 45 | 432 | 30 | 30 | 36 | 3*sqrt(26) | 3*sqrt(26) | 6

%C | 59 | 1440 | 50 | 58 | 72 | sqrt(1394)| sqrt(962)| 2*sqrt(13)

%C | 63 | 480 | 32 | 34 | 34 | 14 | sqrt(130)| sqrt(130)

%C | 80 | 768 | 40 | 40 | 48 | 4*sqrt(26) | 4*sqrt(26) | 8

%C | 83 | 2880 | 74 | 78 | 104 | sqrt(2690)| sqrt(2386)| 2*sqrt(5)

%C .............................................................

%D 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InnerVectenTriangle.html">Inner Vecten Triangle</a>

%e 5 is in the sequence. We use two ways:

%e 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 Second way: by calculation of the sides a', b', c' and by use of Heron's formula.

%e a’ = sqrt((b^2 + c^2 - 4*A)/2) = sqrt((10^2 + 12^2 - 4*48)/2) = sqrt(26);

%e b’ = sqrt((a^2 + c^2 - 4*A)/2) = sqrt((10^2 + 12^2 - 4*48)/2) = sqrt(26);

%e c’ = sqrt((a^2 + b^2 - 4*A)/2) = sqrt((10^2 + 10^2 - 4*48)/2) = 2.

%e Now we use Heron's formula with (a',b',c').

%e We find A' = sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:

%e s1 = (a' + b' + c')/2 = (sqrt(26) + sqrt(26) + 2)/2.

%e We find A' = 5.

%t 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 Cf. A188158, A231275.

%K nonn

%O 1,1

%A _Michel Lagneau_, Nov 06 2013