cp's OEIS Frontend

A370778 Number of non-congruent integer triangles ABC with largest side BC of length n whose opposite vertex A lies inside or on the closed boundary formed by BC and a locus of points A' such that the triangle A'BC has tan A'/2 + tan B/2 + tan C/2 = 2.

%I A370778 #13 Mar 02 2024 13:36:54
%S A370778 0,0,0,0,2,2,3,4,6,7,8,9,13,15,17,19,23,26,28,31,36,39,42,46,51,55,58,
%T A370778 63,70,74,79,83,92,97,102,107,116,121,127,133,143,149,156,163,174,181,
%U A370778 187,195,205,213,220,229,240,248,257,269,279,289,298,306,320,330,340,350,366,375,385,396,412,424,435,448,462,474,487
%N A370778 Number of non-congruent integer triangles ABC with largest side BC of length n whose opposite vertex A lies inside or on the closed boundary formed by BC and a locus of points A' such that the triangle A'BC has tan A'/2 + tan B/2 + tan C/2 = 2.
%C A370778 The boundary formed by BC and a locus of points A' such that triangle A'BC has tan A'/2 + tan B/2 + tan C/2 = 2 approximates a semi-ellipse with major axis BC = n and minor axis 3n/4. See Soddyian triangle link below for exact curve formula.
%C A370778 Integer triangles ABC with semitangents tan A/2 + tan B/2 + tan C/2 = 2 that lie on the boundary have outer Soddy circles that have degenerated into straight lines. Such triangles are Heronian and their sides a <= b <= c are constrained by 1/sqrt(s-c) = 1/sqrt(s-b) + 1/sqrt(s-a) where s is the triangle's semiperimeter. This sequence is the number of integer triangles with base length BC = n such that their semitangents tan A/2 + tan B/2 + tan C/2 >= 2. Integer triangles within the boundary have outer Soddy circles with negative curvature. Those outside the boundary have outer Soddy circles with positive curvature. Those that lie on the boundary have outer Soddy circles with zero curvature, i.e., their outer Soddy circles have degenerated into straight lines.
%C A370778 Furthermore, "An integer triangle ABC has exactly one isoperimetric point J1 and exactly one point of equal detour J2 if and only if tan A/2 + tan B/2 + tan C/2 < 2; it has exactly two equal detour points D1 and D2 and no isoperimetric point if and only if tan A/2 + tan B/2 + tan C/2 > 2; it has exactly one equal detour point D and no isoperimetric point if and only if tan A/2 + tan B/2 + tan C/2 = 2. In the first case, J1 and J2 coincide with the centers of the outer and inner Soddy circles, respectively. In the second case, D1 and D2 are the centers of the two Soddy circles. In the third case, D is the center of the inner Soddy circle, while the outer Soddy circle is a straight line." (Quoted from Hajja, Yff paper - see link below.)
%H A370778 Muwaffaq Hajja and Peter Yff, <a href="https://www.researchgate.net/publication/225560690_The_isoperimetric_point_and_the_points_of_equal_detour_in_a_triangle">The isoperimetric point and the point(s) of equal detour in a triangle</a>, J. geom. 87, 76-82 (2007).
%H A370778 Frank M. Jackson, <a href="https://forumgeom.fau.edu/FG2013volume13/FG201301index.html">Soddyian triangles</a>, Forum Geom., 13 (2013), 1-6.
%H A370778 Frank M. Jackson and Stalislav Takhaev, <a href="https://forumgeom.fau.edu/FG2015volume15/FG201502index.html">Heronian Triangles of Class K: Congruent Incircles Cevian Perspective</a>, Forum Geom., 15 (2015) 5-12.
%H A370778 Wikipedia, <a href="https://en.wikipedia.org/wiki/Soddy_circles_of_a_triangle">Soddy circles of a triangle</a>.
%e A370778 a(8)=4 as there are 3 non-congruent integer triangles with base length of 8 whose apex lies inside the boundary and one non-congruent integer triangle that lies on the boundary. The integer triples are (4, 5, 8), (5, 5, 8), (3, 6, 8), (2, 7, 8) with (5, 5, 8) lying on the boundary. There are 16 other triangles from the complete set of non-congruent integer triangles with largest side length 8 (A002620(8+1)) = 20 that are outside the boundary.
%t A370778 striangles[c_] :=  Module[{lst={}, a, b, s, A, ta, tb, tc}, Do[If[a+b>c, (s=(a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; ta=A/(s(s-a)); tb=A/(s(s-b)); tc=A/(s(s-c)); If[ta+tb+tc>=2, AppendTo[lst, {a, b, c}]])], {b, 1, c}, {a, 1, b}]; lst]; Table[Length@striangles[n], {n, 1, 100}]
%Y A370778 Cf. A002620, A236384, A251607.
%K A370778 nonn
%O A370778 1,5
%A A370778 _Frank M Jackson_, Mar 01 2024