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

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, 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, 8, 8, 7, 1, 8, 6, 7, 2, 0, 6, 9, 8, 5, 6, 2, 1, 8, 0

Offset: 1

Author

Clark Kimberling, Nov 13 2024

Keywords

• nonn
• cons

Comments

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,

b = length of shortest side = 1.1891197542670080135...

c = length of longest side = 1.70109767501680105234...

a = length of other side = 1.9240361790979417706848...

area = 1

perimeter = a+b+c = 4.8142536083817508366273974008...

circumradius = 0.972989352363244654532817794159950...

inradius = 0.4154330375362743229952970705418968790...

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).

Triangle center NTC NBC

incenter (0.6366, 0.6366, 0.6366) (0.4521, 0.2092, 0.3385)

centroid (0.4693, 1.0140, 0.6267) (1/3, 1/3, 1/3)

circumcenter (0.2396, 1.2668, 0.7772) (0.1702, 0.4164, 0.4133)

orthocenter (1.0678, 0.2020, 0.3292) (0.7196, 0.0841, 0.1961)

nine-point center (0.0696, 0.0710, 1.7432) (0.4299, 0.2710, 0.2990)

Examples

			1.189119754267008013592915705505374859646258022049...
		

Crossrefs

Cf. A001622, A152149.

Programs

Mathematica

r = (1 + 5^(1/2))/2;
b = FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]];
c = r*b ; (* angle C, where b = angle B *)
a = Pi - r^2 b; (* angle A *)
{a1, b1, c1} = {Sin[a], Sin[b], Sin[c]}
k = 2/((a1 + b1 + c1) (-a1 + b1 + c1) (-b1 + c1 + a1) (-c1 + a1 + b1))^(1/4)
{k a1, k b1, k c1} (* sidelengths *)
k*b1 (* length of shortest side *)
RealDigits[k b1][[1]] (* this sequence *)