X3067
HOFSTADTER ELLIPSE INTERSECTION
The Hofstadter ellipse E(0) is described at
X359 . The point other than
A ,
B , and
C in which E(0) meets the circumcircle is
X3067 . As with
X359 and
X360 , this is a
transcendental center , with “exposed angles”
A ,
B ,
C in its coordinates.
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3067 = HOFSTADTER ELLIPSE INTERSECTION (0.2031, 0.5171)
A
B
C
X3067
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3067 = HOFSTADTER ELLIPSE INTERSECTION (0.2919, 0.4547)
A
B
C
X3067
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3067 = HOFSTADTER ELLIPSE INTERSECTION (0.3226, 0.4675)
A
B
C
X3067
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3067 = HOFSTADTER ELLIPSE INTERSECTION (0.4342, 0.2920)
A
B
C
X3067
Kimberling (6, 9, 13) | AB = 1
X3067 (HOFSTADTER ELLIPSE INTERSECTION ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a/(AB - AC) : b/(BC - BA) : c/(CA -CB)\)
Barycentrics: \(a^{2}/(AB - AC) : b^{2}/(BC - BA) : c^{2}/(CA -CB)\)
Search Value (6, 9, 13): \(1.445571013027\)
