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A343891 List of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

%I A343891 #24 May 05 2021 21:05:14
%S A343891 4,3,6,12,10,15,15,12,20,21,15,35,24,21,28,35,30,42,40,36,45,45,35,63,
%T A343891 55,40,88,56,44,77,60,55,66,63,56,72,72,52,117,77,63,99,80,65,104,84,
%U A343891 78,91,91,70,130,99,90,110,105,77,165,112,105,120,117,99,143,120,85,204,132,102,187
%N A343891 List of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.
%C A343891 The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
%C A343891 When sides satisfy 2/a = 1/b + 1/c, or a = 2*b*c/(b+c) then a is always the middle side with b < a < c.
%C A343891 Equivalent relations: the heights and sines satisfy 2*h_a = h_b + h_c and 2/sin(A) = 1/sin(B) + 1/sin(C).
%C A343891 Inequalities between sides: a/2 < b < a < c < b*(1+sqrt(2)).
%D A343891 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-337 p. 179, André Desvigne.
%e A343891 (4, 3, 6) is the first triple with 2/4 = 1/3 + 1/6 and 6-4 < 3 < 6+4.
%e A343891 The table begins:
%e A343891    4,  3,  6;
%e A343891   12, 10, 15;
%e A343891   15, 12, 20;
%e A343891   21, 15, 35;
%e A343891   24, 21, 28;
%e A343891   35, 30, 42;
%e A343891   ...
%p A343891 for a from 4 to 200 do
%p A343891 for b from floor(a/2)+1 to a-1 do
%p A343891 c := a*b/(2*b-a);
%p A343891 if c=floor(c) and igcd(a,b,c)=1 and c-b<a then print(a,b,c); end if;
%p A343891 end do;
%p A343891 end do;
%Y A343891 Cf. A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).
%K A343891 nonn,tabf
%O A343891 1,1
%A A343891 _Bernard Schott_, May 03 2021