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A376961 Length of the shortest side of the doubly golden triangle (A152149) that has area 1.

%I A376961 #4 Nov 17 2024 07:31:53
%S A376961 1,1,8,9,1,1,9,7,5,4,2,6,7,0,0,8,0,1,3,5,9,2,9,1,5,7,0,5,5,0,5,3,7,4,
%T A376961 8,5,9,6,4,6,2,5,8,0,2,2,0,4,9,3,6,0,5,6,4,9,5,4,1,8,0,2,0,9,1,2,2,5,
%U A376961 8,8,7,1,8,6,7,2,0,6,9,8,5,6,2,1,8,0
%N A376961 Length of the shortest side of the doubly golden triangle (A152149) that has area 1.
%C A376961 The unique (shape of) triangle ABC that is both angle-golden and side-golden is discussed in A152149. The vertex angles, A,B,C are unique with A = B*tau and C = Pi - C*tau^2, where tau = golden ratio (A001622), but the lengths a,b,c of sides opposite A,B,C are not unique.  Instead, they are proportional to sin A, sin B, sin C. Consequently, if ABC is scaled so that its area is 1, then the sidelengths are unique. In that case,
%C A376961 b = length of shortest side = 1.1891197542670080135...
%C A376961 c = length of longest side = 1.70109767501680105234...
%C A376961 a = length of other side = 1.9240361790979417706848...
%C A376961 area = 1
%C A376961 perimeter = a+b+c = 4.8142536083817508366273974008...
%C A376961 circumradius = 0.972989352363244654532817794159950...
%C A376961 inradius = 0.4154330375362743229952970705418968790...
%C A376961 The following list gives approximate coordinates for five well-known triangle centers in the plane of the doubly golden triangle that has area 1. Approximate distances from each of these points X to the sidelines BC, CA, AB appear under the heading NTC (normalized trilinear coordinates), and approximate areas of the triangles BXC, CXA, AXB appear under NBC (normalized barycentric coordinates).
%C A376961 Triangle center     NTC                          NBC
%C A376961 incenter            (0.6366, 0.6366, 0.6366)   (0.4521, 0.2092, 0.3385)
%C A376961 centroid            (0.4693, 1.0140, 0.6267)   (1/3, 1/3, 1/3)
%C A376961 circumcenter        (0.2396, 1.2668, 0.7772)   (0.1702, 0.4164, 0.4133)
%C A376961 orthocenter         (1.0678, 0.2020, 0.3292)   (0.7196, 0.0841, 0.1961)
%C A376961 nine-point center   (0.0696, 0.0710, 1.7432)   (0.4299, 0.2710, 0.2990)
%e A376961 1.189119754267008013592915705505374859646258022049...
%t A376961 r = (1 + 5^(1/2))/2;
%t A376961 b = FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]];
%t A376961 c = r*b ; (* angle C, where b = angle B *)
%t A376961 a = Pi - r^2 b; (* angle A *)
%t A376961 {a1, b1, c1} = {Sin[a], Sin[b], Sin[c]}
%t A376961 k = 2/((a1 + b1 + c1) (-a1 + b1 + c1) (-b1 + c1 + a1) (-c1 + a1 + b1))^(1/4)
%t A376961 {k a1, k b1, k c1} (* sidelengths *)
%t A376961 k*b1 (* length of shortest side *)
%t A376961 RealDigits[k b1][[1]] (* this sequence *)
%Y A376961 Cf. A001622, A152149.
%K A376961 nonn,cons
%O A376961 1,3
%A A376961 _Clark Kimberling_, Nov 13 2024