OFFSET
1,3
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)
EXAMPLE
1.189119754267008013592915705505374859646258022049...
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 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 13 2024
STATUS
approved