X545
DIRECTION OF VECTOR AX + BX + CX, where X = X190
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, 45 }: Central Line L(2, 45) { 30, 511 }: Central Line L(30, 511) { 44, 1266 }: Central Line L(44, 1266) { 321, 1227 }: Central Line L(321, 1227)
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_45 = -BETH CONJUGATE OF X(1) (0.5418, 0.1981) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_1266 = ISOTOMIC CONJUGATE OF X(1120) (2.8367, 1.7996) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4800, 0.1527) X_1227 = ISOTOMIC CONJUGATE OF X(1168) (0.2551, 0.2499) X_545 = DIRECTION OF VECTOR AX + BX + CX, where X = X(190) (-0.0013, 0.4328)
A
B
C
X2
X45
X30
X511
X44
X1266
X321
X1227
X545
(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, 45 }: Central Line L(2, 45) { 30, 511 }: Central Line L(30, 511) { 44, 1266 }: Central Line L(44, 1266) { 321, 1227 }: Central Line L(321, 1227)
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_45 = -BETH CONJUGATE OF X(1) (0.5565, 0.1565) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2569, -0.3367) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_1266 = ISOTOMIC CONJUGATE OF X(1120) (2.2800, 0.9600) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4525, 0.1050) X_1227 = ISOTOMIC CONJUGATE OF X(1168) (0.2400, 0.1800) X_545 = DIRECTION OF VECTOR AX + BX + CX, where X = X(190) (0.0044, 0.3514)
A
B
C
X2
X45
X30
X511
X44
X1266
X321
X1227
X545
(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, 45 }: Central Line L(2, 45) { 30, 511 }: Central Line L(30, 511) { 44, 1266 }: Central Line L(44, 1266) { 321, 1227 }: Central Line L(321, 1227)
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_45 = -BETH CONJUGATE OF X(1) (0.5474, 0.1580) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2881, -0.3552) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_1266 = ISOTOMIC CONJUGATE OF X(1120) (2.1219, 1.0180) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4606, 0.1074) X_1227 = ISOTOMIC CONJUGATE OF X(1168) (0.2683, 0.2014) X_545 = DIRECTION OF VECTOR AX + BX + CX, where X = X(190) (0.0227, 0.4144)
A
B
C
X2
X45
X30
X511
X44
X1266
X321
X1227
X545
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, 45 }: Central Line L(2, 45) { 30, 511 }: Central Line L(30, 511) { 44, 1266 }: Central Line L(44, 1266) { 321, 1227 }: Central Line L(321, 1227)
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_45 = -BETH CONJUGATE OF X(1) (0.5647, 0.0872) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_511 = ISOGONAL CONJUGATE OF X(98) (0.3235, -0.5244) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_1266 = ISOTOMIC CONJUGATE OF X(1120) (1.6862, 0.3633) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4100, 0.0466) X_1227 = ISOTOMIC CONJUGATE OF X(1168) (0.2413, 0.0979) X_545 = DIRECTION OF VECTOR AX + BX + CX, where X = X(190) (-0.0832, 0.2844)
A
B
C
X2
X45
X30
X511
X44
X1266
X321
X1227
X545
Kimberling (6, 9, 13) | AB = 1
X545 (DIRECTION OF VECTOR AX + BX + CX, where X = X190 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where \(f(a,b,c) = (2ax - by - cz)/a\) , \(x = x(a,b,c) = bc/(b - c)\) , \(y = x(b,c,a)\) , \(z = x(c,a,b)\)
Barycentrics: \(2 a^{2} - b^{2} - c^{2} - 2 a b - 2 a c + 4 b c : :\)
X545
▸ Incident Central Lines5 lines
{2 45 30 511 44 1266 63 2161 321 1227
▸ Functional Relations5 relations
