X3138
X103 -CEVA CONJUGATE OF X523
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 { 116, 125 }: Central Line L(116, 125) { 2972, 2973 }: Central Line L(2972, 2973)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_116 = MIDPOINT OF X(4) AND X(103) (0.6990, 0.0375) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_2972 = 6th MACBEATH POINT (0.5552, 0.0030) X_2973 = 7th MACBEATH POINT (0.7494, 0.1928) X_3138 = -CEVA CONJUGATE OF X(523) (0.4066, -0.1423)
A
B
C
X103
X523
X2
X3
X116
X125
X2972
X2973
X3138
(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 { 116, 125 }: Central Line L(116, 125) { 2972, 2973 }: Euler Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_103 = ANTIPODE OF X(101) (0.6923, -0.4615) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0987, -0.0010) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_116 = MIDPOINT OF X(4) AND X(103) (0.6662, 0.0092) X_125 = CENTER OF JERABEK HYPERBOLA (0.6400, 0.0000) X_2972 = 6th MACBEATH POINT (0.5000, 0.0000) X_2973 = 7th MACBEATH POINT (0.6400, 0.4800) X_3138 = -CEVA CONJUGATE OF X(523) (0.4839, -0.0551)
A
B
C
X103
X523
X2
X3
X116
X125
X2972
X2973
X3138
(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 { 116, 125 }: Central Line L(116, 125) { 2972, 2973 }: Euler Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_103 = ANTIPODE OF X(101) (0.6635, -0.4725) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0981, 0.0262) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_116 = MIDPOINT OF X(4) AND X(103) (0.6408, 0.0067) X_125 = CENTER OF JERABEK HYPERBOLA (0.6180, 0.0000) X_2972 = 6th MACBEATH POINT (0.5000, 0.0000) X_2973 = 7th MACBEATH POINT (0.6180, 0.4859) X_3138 = -CEVA CONJUGATE OF X(523) (0.4909, -0.0373)
A
B
C
X103
X523
X2
X3
X116
X125
X2972
X2973
X3138
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 { 116, 125 }: Central Line L(116, 125) { 2972, 2973 }: Central Line L(2972, 2973)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_103 = ANTIPODE OF X(101) (0.5518, -0.8428) X_523 = ISOGONAL CONJUGATE OF X(110) (1.1957, 0.0148) X_2 = CENTROID (0.5444, 0.0934) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_116 = MIDPOINT OF X(4) AND X(103) (0.5925, -0.0067) X_125 = CENTER OF JERABEK HYPERBOLA (0.5678, -0.0079) X_2972 = 6th MACBEATH POINT (0.4988, -0.0025) X_2973 = 7th MACBEATH POINT (0.7132, -0.0484) X_3138 = -CEVA CONJUGATE OF X(523) (0.5320, -0.0096)
A
B
C
X103
X523
X2
X3
X116
X125
X2972
X2973
X3138
Kimberling (6, 9, 13) | AB = 1
X3138 (X103 -CEVA CONJUGATE OF X523 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\) , where
\(x : y : z\) =
X103 and
\(u : v : w\) =
X523 Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\) , where \(f(a,b,c) = u(-u/x + v/y + w/z)\)
Search Value (6, 9, 13): \(3.790962809108\)
▸ Incident Central Lines3 lines
{2 3 116 125 2972 2973
▸ Functional Relations1 relation
Notes
As a point on the Euler line, X3138 \((\$aS_{A}^{5}\$ -\$aS_{A}^{2}\$[(E+F)^{3}-3ES^{2}] +\$aS_{B}S_{C}\$ES^{2} +\$aS_{A}\$(E+F)FS^{2}-\$a\$(E+F)^{2}FS^{2}, -\$aS_{A}^{3}\$S^{2} +\$aS_{A}^{2}\$(E+F)S^{2} -\$aS_{B}S_{C}\$(E+F)S^{2}+\$a\$FS^{4})\) .
