A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.

3, 5, 7, 7, 8, 13, 5, 16, 19, 11, 24, 31, 7, 33, 37, 13, 35, 43, 16, 39, 49, 9, 56, 61, 32, 45, 67, 17, 63, 73, 40, 51, 79, 11, 85, 91, 19, 80, 91, 55, 57, 97, 40, 77, 103, 24, 95, 109, 13, 120, 127, 23, 120, 133, 65, 88, 133, 69, 91, 139, 56, 115, 151, 25, 143, 157, 75, 112, 163, 15, 161, 169, 104, 105, 181

Offset: 1

Author

Bernard Schott, Sep 22 2022

Keywords

• nonn
• tabf

Comments

The only triangles with integer sides that have an angle equal to a whole number of degrees are triangles which have an angle of 60° (A335893), or an angle of 90° (A263728) or an angle of 120° as here (see Keith Selkirk link, p. 251).

The triples are displayed in nondecreasing order of largest side c, and if largest sides coincide then by increasing order of the smallest side a, hence, each triple (a, b, c) is in increasing order.

The corresponding metric relation between sides is c^2 = a^2 + a*b + b^2.

The triples (a, b, c) can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u:

-> a = u^2 - v^2

-> b = 2*u*v + v^2

-> c = u^2 + u*v + v^2.

Note that side c cannot be even when the triple is primitive as here.

The (3, 5, 7) triangle is the only primitive triangle with a 120-degree angle and with its integer sides in arithmetic progression (A336750). This smallest triple is obtained for u = 2 and v = 1.

The Fermat point of these triangles is vertex C, then distance FA+FB+FC = CA+CB = b+a is an integer.

If (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893, see Emrys Read link, lemma 2 p. 302.

Examples

			Table of triples begins:
   3,  5,  7;
   7,  8, 13;
   5, 16, 19;
  11, 24, 31;
   7, 33, 37;
............
(7, 8, 13) is a triple for this sequence because from the law of cosines (see link), cos(C) = (7^2 + 8^2 - 13^2)/(2*7*8) = -1/2.
		

Links

• G. Julia, Triangles dont un angle mesure 120 degrés, Problème Capes (in French).
• Emrys Read, On integer-sided triangles containing angles of 120° or 60°, Mathematical Gazette 90, July 2006, 299-305.
• Keith Selkirk, Integer-Sided Triangles with an Angle of 120°, Mathematical Gazette, Vol. 67, No. 442 (Dec., 1983), pp. 251-255.
• Eric Weisstein's World of Mathematics, Law of Cosines.
• Wikipedia, Triangles with an angle of 120°, Eisenstein triple.

Crossrefs

Cf. A357275, A357276, A357277, A357278.

Cf. also A263728, A336750, A335893 (similar with an angle of Pi/3).

Programs

Maple

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a
				

Extensions

a(31..33) = 40,51,79 inserted by Georg Fischer, Dec 04 2022