login
A371969
Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.
1
49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240
OFFSET
1,1
REFERENCES
These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.
LINKS
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
EXAMPLE
a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].
PROG
(PARI) H(a, b, c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1, w2, w3, v1, v2, v3) = matdet([0, 1, 1, 1, 1; 1, 0, w3^2, w2^2, v1^2; 1, w3^2, 0, w1^2, v2^2; 1, w2^2, w1^2, 0, v3^2; 1, v1^2, v2^2, v3^2, 0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1], w[2], w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3], v2, v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1], v2, v3)); if (A1>=A, next); my (A2=H(w[2], v1, v3)); if (A2>=A, next); my (C=CM(w[1], w[2], w[3], v1, v2, v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri, " ", Vec(w), " ", [v1, v2, v3]);
return(1))))), [1, (peri-1)\2], [3, 3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n, ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Nagel and Hugo Pfoertner, Apr 14 2024
STATUS
approved