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A333391 Longest side of primitive integer triangles with nonzero rational distances between three vertices and first isogonic center, sorted.

%I A333391 #26 Mar 02 2025 13:03:49
%S A333391 73,95,152,205,208,280,285,287,296,343,361,387,407,437,469,473,485,
%T A333391 497,507,608,624,633,645,713,715,728,728,817,873,931,1016,1273,1288,
%U A333391 1311,1313,1343,1368,1387,1443,1457,1463,1469,1477,1488,1519,1519,1560,1584,1591,1591
%N A333391 Longest side of primitive integer triangles with nonzero rational distances between three vertices and first isogonic center, sorted.
%H A333391 Leisure Maths Entertainment Forum, <a href="https://kuing.cjhb.site/thread-6994-1-1.html">The primitive integer triangles with nonzero rational distances between three vertices and 1st isogonic center</a>, Chinese blog.
%H A333391 Project Euler, <a href="https://projecteuler.net/problem=143">Problem 143. Investigating the Torricelli point of a triangle</a>
%e A333391 Case 1: When the isogonic center is inside the triangle, i.e., the three internal angles are all less than 120 degrees. Example: Length of three sides (a, b, c) = (57, 65, 73), rational distances with signs (x, y, z) = (325/7, 264/7, 195/7);
%e A333391 Case 2: When the isogonic center is outside the triangle, i.e., an internal angle is greater than 120 degrees. Example: Lengths of three sides (a, b, c) = (43, 248, 285), rational distances with signs (x, y, z) = (1800/7, 345/7, -136/7);
%e A333391 Thus 73 and 285 are in this sequence.
%e A333391 a(26) = a(27) = 728 is the smallest longest side that appears twice because: (a, b, c) = (57, 673, 728) is a triple with (x, y, z) = (9016/13, 840/13, -561/13), and (a, b, c) = (403, 725, 728) is a triple with (x, y, z) = (203000/349, 81928/349, 80475/349). - _Jinyuan Wang_, Feb 12 2025
%o A333391 (PARI) lista(nn) = my(d); for(c=4, nn, for(b=(c+2)\2, c-1, for(a=c-b+1, b-1, if(gcd([a, b, c])==1 && a^2+b^2+a*b!=c^2 && issquare(6*(a^2*b^2+b^2*c^2+c^2*a^2)-3*(a^4+b^4+c^4), &d) && issquare((a^2+b^2+c^2+d)/2), print1(c, ", "))))); \\ _Jinyuan Wang_, Feb 12 2025
%Y A333391 Cf. A070082, A336332.
%K A333391 nonn
%O A333391 1,1
%A A333391 _Mo Li_, Mar 18 2020
%E A333391 More terms from _Jinyuan Wang_, Feb 12 2025