X562
TRILINEAR QUOTIENT X2 *X50 /X49
X562 = polar conjugate of X
30529
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) { 4, 93 }: Central Line L(4, 93)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_93 = ISOGONAL CONJUGATE OF X(49) (0.5299, 0.2878) X_562 = TRILINEAR QUOTIENT X(2)*X(50)/X(49) (3.1429, 3.7154)
A
B
C
X4
X93
X562
(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) { 2, 3 }: Euler Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_562 = TRILINEAR QUOTIENT X(2)*X(50)/X(49) (0.6400, 0.4800)
A
B
C
X2
X3
X562
(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) { 2, 3 }: Euler Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_562 = TRILINEAR QUOTIENT X(2)*X(50)/X(49) (0.6180, 0.4859)
A
B
C
X2
X3
X562
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) { 4, 93 }: Central Line L(4, 93)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_4 = ORTHOCENTER (0.6331, 0.8294) X_93 = ISOGONAL CONJUGATE OF X(49) (-1.9471, 0.4523) X_562 = TRILINEAR QUOTIENT X(2)*X(50)/X(49) (-5.3006, -0.0378)
A
B
C
X4
X93
X562
Kimberling (6, 9, 13) | AB = 1
X562 (TRILINEAR QUOTIENT X2 *X50 /X49 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\csc A \tan 3A : \csc B \tan 3B : \csc C \tan 3C\)
Barycentrics: \(\tan 3A : \tan 3B : \tan 3C\)
Search Value (6, 9, 13): \(50.090809847426\)
▸ Linear Combinations1 combination
▸ Incident Central Lines2 lines
▸ Functional Relations1 relation
Notes
Clark Kimberling and Cyril Parry, “Products, Square Roots, and Layers in Triangle Geometry,” Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
