X1982
PU(22)-HARMONIC CONJUGATE OF X1981
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) { 2, 3 }: Euler Line { 1, 648 }: Central Line L(1, 648) { 255, 1098 }: Central Line L(255, 1098)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1 = INCENTER (0.5714, 0.2333) X_648 = TRILINEAR POLE OF EULER LINE (0.4563, 0.5855) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4217, 0.0365) X_1098 = TRILINEAR SQUARE OF X(21) (0.4785, 0.1286) X_1982 = PU(22)-HARMONIC CONJUGATE OF X(1981) (0.5616, 0.2633)
A
B
C
X2
X3
X4
X1
X648
X255
X1098
X1982
(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 { 1, 648 }: Incenter–Orthocenter Line { 255, 1098 }: Central Line L(255, 1098)
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_4 = ORTHOCENTER (0.6400, 0.4800) X_1 = INCENTER (0.6000, 0.2000) X_648 = TRILINEAR POLE OF EULER LINE (0.6400, 0.4800) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4286, 0.0000) X_1098 = TRILINEAR SQUARE OF X(21) (0.4600, 0.0715) X_1982 = PU(22)-HARMONIC CONJUGATE OF X(1981) (0.6400, 0.4800)
A
B
C
X2
X3
X4
X1
X648
X255
X1098
X1982
(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 { 1, 648 }: Incenter–Orthocenter Line { 255, 1098 }: Central Line L(255, 1098)
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_4 = ORTHOCENTER (0.6180, 0.4859) X_1 = INCENTER (0.5841, 0.2021) X_648 = TRILINEAR POLE OF EULER LINE (0.6180, 0.4859) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4401, 0.0000) X_1098 = TRILINEAR SQUARE OF X(21) (0.4667, 0.0725) X_1982 = PU(22)-HARMONIC CONJUGATE OF X(1981) (0.6180, 0.4859)
A
B
C
X2
X3
X4
X1
X648
X255
X1098
X1982
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) { 2, 3 }: Euler Line { 1, 648 }: Central Line L(1, 648) { 255, 1098 }: Central Line L(255, 1098)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2 = CENTROID (0.5444, 0.0934) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_4 = ORTHOCENTER (0.6331, 0.8294) X_1 = INCENTER (0.6154, 0.1300) X_648 = TRILINEAR POLE OF EULER LINE (0.7616, 0.2752) X_255 = ISOGONAL CONJUGATE OF X(158) (0.5536, 0.0615) X_1098 = TRILINEAR SQUARE OF X(21) (0.4475, 0.0110) X_1982 = PU(22)-HARMONIC CONJUGATE OF X(1981) (0.5397, 0.0549)
A
B
C
X2
X3
X4
X1
X648
X255
X1098
X1982
Kimberling (6, 9, 13) | AB = 1
X1982 (PU(22)-HARMONIC CONJUGATE OF X1981 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((2x - y - z)/x : :\) , where \(x = x(a,b,c) = \sin 2A \sin(B - C)\)
Search Value (6, 9, 13): \(3.063044308111\)
▸ Incident Central Lines3 lines
{1 648 2 3 255 1098
▸ Functional Relations2 relations
Notes
As a point on the Euler line, X1982 \((2\$aS_{A}^{2}S_{B}\$F -2\$aS_{C}S_{A}^{2}\$F +\$aS_{C}^{2}S_{A}\$F -\$aS_{A}S_{B}^{2}\$F, \$aS_{C}S_{A}\$S^{2} -\$aS_{A}S_{B}\$S^{2}+3\$aS_{B}\$FS^{2} -3\$aS_{C}\$FS^{2})\) .
