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A335893 Primitive triples for integer-sided triangles whose angles A < B < C are in arithmetic progression.

%I A335893 #64 Sep 21 2022 12:01:06
%S A335893 3,7,8,5,7,8,7,13,15,8,13,15,5,19,21,16,19,21,11,31,35,24,31,35,7,37,
%T A335893 40,33,37,40,13,43,48,35,43,48,16,49,55,39,49,55,9,61,65,56,61,65,32,
%U A335893 67,77,45,67,77,17,73,80,63,73,80,40,79,91,51,79,91,11,91,96
%N A335893 Primitive triples for integer-sided triangles whose angles A < B < C are in arithmetic progression.
%C A335893 The triples are displayed in nondecreasing order of middle side, and if middle sides coincide then by increasing order of the largest side, hence, each triple (a, b, c) is in increasing order.
%C A335893 These three properties below are equivalent:
%C A335893 -> integer-sided triangles whose angles A < B < C are in arithmetic progression,
%C A335893 -> integer-sided triangles such that B = (A+C)/2 with A < C,
%C A335893 -> integer-sided triangles such that A < B < C with B = Pi/3.
%C A335893 When A < B < C are in arithmetic progression with B = A + phi and C = B + phi, then 0 < phi < Pi/3.
%C A335893 The corresponding metric relation between sides is b^2 = a^2 - a*c + c^2.
%C A335893 There exists such primitive triangle iff b^2 is an odd square term of A024612. Hence, the first few middle sides b are 7, 13, 19, 31, 37, 43, 49, 61, 67, ... and b is a term of A004611 \ {1}. Indeed, b cannot be even if the triple is primitive.
%C A335893 As B = Pi/3 and C runs from Pi/3 to 2*Pi/3, sin(C) gets a maximum when C = Pi/2 with sin(C) = 1, hence, from law of sines (see link): b/sin(B) = c/sin(C), and c < b/sin(Pi/3) = b * 2/sqrt(3) < 6*b/5. This bound is used in the PARI and Maple programs below.
%C A335893 When triple (a, b, c) is solution, then triple (c-a, b, c) is another solution. Hence, for each b odd solution, there exist 2 triples with same middle side b and same largest side c.
%C A335893 The common tangent to the nine-point circle and the incircle of a triangle ABC is parallel to the Euler line iff angles A < B < C are in arithmetic progression (see Crux Mathematicorum for Indian team selection). - _Bernard Schott_, Apr 14 2022
%C A335893 These triples are called (primitive) Eisenstein triples (Wikipedia). - _Bernard Schott_, Sep 21 2022
%D A335893 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.
%H A335893 Bill Sands, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv36n5.pdf">Indian team selection test 2007, question 5</a>, Crux Mathematicorum, Vol. 36, No. 5 (2010), p. 278.
%H A335893 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LawofCosines.html">Law of Cosines</a>.
%H A335893 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LawofSines.html">Law of Sines</a>.
%H A335893 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_triple">Eisenstein triple</a>.
%H A335893 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%e A335893 (3, 7, 8) is a triple for this sequence because from law of cosines (see link), cos(A) = (7^2 + 8^2 - 3^2)/(2*7*8) = 13/14, cos(B) = (8^2 + 3^2 - 7^2)/(2*8*3) = 1/2 and cos(C) = (3^2 + 7^2 - 8^2)/(2*3*7) = -1/7; then, (A+C)/2 = ( arccos(13/14) + arccos(-1/7) )/2 = Pi/3 = B.
%e A335893 Also, arccos(13/14) ~ 21.787 degrees, arccos(1/2) = 60 degrees, arccos(-1/7) ~ 98.213 degrees, so B-A = C-B ~ 38.213 degrees, hence (A, B, C) are in arithmetic progression.
%e A335893 5^2 - 5*8 + 8^2 = 7^2, hence (5, 7, 8) is another triple for triangle whose angles A < B < C are in arithmetic progression.
%p A335893 for b from 3 to 250 by 2 do
%p A335893 for c from b+1 to 6*b/5 do
%p A335893 a := (c - sqrt(4*b^2-3*c^2))/2;
%p A335893 if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(a,b,c,c-a,b,c); end if;
%p A335893 end do;
%p A335893 end do;
%o A335893 (PARI) lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print(a, ", ", b, ", ", c, ", "); print(c-a, ", ", b, ", ", c, ", ");););););} \\ _Michel Marcus_, Jul 15 2020
%Y A335893 Cf. A004611, A024612.
%Y A335893 Cf. A335894 (smallest side), A335895 (middle side), A335896 (largest side), A335897 (perimeter).
%Y A335893 Cf. A103606 (primitive Pythagorean triples), A335034 (primitive triples for triangles with two perpendicular medians).
%K A335893 nonn,tabf
%O A335893 1,1
%A A335893 _Bernard Schott_, Jun 29 2020