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

73, 95, 152, 205, 208, 280, 285, 287, 296, 343, 361, 387, 407, 437, 469, 473, 485, 497, 507, 608, 624, 633, 645, 713, 715, 728, 728, 817, 873, 931, 1016, 1273, 1288, 1311, 1313, 1343, 1368, 1387, 1443, 1457, 1463, 1469, 1477, 1488, 1519, 1519, 1560, 1584, 1591, 1591

Offset: 1

Author

Mo Li, Mar 18 2020

Keywords

• nonn

Examples

			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);
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);
Thus 73 and 285 are in this sequence.
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
		

Links

• Leisure Maths Entertainment Forum, The primitive integer triangles with nonzero rational distances between three vertices and 1st isogonic center, Chinese blog.
• Project Euler, Problem 143. Investigating the Torricelli point of a triangle

Crossrefs

Cf. A070082, A336332.

Programs

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

Extensions

More terms from Jinyuan Wang, Feb 12 2025