X3137
X102 -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 { 124, 125 }: Central Line L(124, 125)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_102 = Λ(INCENTER, ORTHOCENTER) (0.6364, -0.3897) 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_124 = -OF-MEDIAL-TRIANGLE (0.6243, 0.0030) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_3137 = -CEVA CONJUGATE OF X(523) (0.4320, -0.0759)
A
B
C
X102
X523
X2
X3
X124
X125
X3137
(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 { 124, 125 }: Central Line L(124, 125)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_102 = Λ(INCENTER, ORTHOCENTER) (0.5000, -0.5000) 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_124 = -OF-MEDIAL-TRIANGLE (0.5700, -0.0100) X_125 = CENTER OF JERABEK HYPERBOLA (0.6400, 0.0000) X_3137 = -CEVA CONJUGATE OF X(523) (0.4939, -0.0209)
A
B
C
X102
X523
X2
X3
X124
X125
X3137
(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 { 124, 125 }: Central Line L(124, 125)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_102 = Λ(INCENTER, ORTHOCENTER) (0.5000, -0.5000) 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_124 = -OF-MEDIAL-TRIANGLE (0.5590, -0.0071) X_125 = CENTER OF JERABEK HYPERBOLA (0.6180, 0.0000) X_3137 = -CEVA CONJUGATE OF X(523) (0.4965, -0.0146)
A
B
C
X102
X523
X2
X3
X124
X125
X3137
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 { 124, 125 }: Central Line L(124, 125)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_102 = Λ(INCENTER, ORTHOCENTER) (0.3952, -0.8354) 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_124 = -OF-MEDIAL-TRIANGLE (0.5142, -0.0030) X_125 = CENTER OF JERABEK HYPERBOLA (0.5678, -0.0079) X_3137 = -CEVA CONJUGATE OF X(523) (0.5326, -0.0047)
A
B
C
X102
X523
X2
X3
X124
X125
X3137
Kimberling (6, 9, 13) | AB = 1
X3137 (X102 -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
\(x : y : z\) =
X102 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): \(3.735634957834\)
▸ Incident Central Lines2 lines
▸ Functional Relations1 relation
Notes
As a point on the Euler line, X3137 \((\$aS_{A}^{6}\$ -\$aS_{A}^{3}\$[(E+F)^{3}-3ES^{2}] +\$aS_{B}S_{C}\$(E+F)FS^{2}-\$a\$F^{2}S^{4}, -\$aS_{A}^{4}\$S^{2} +\$aS_{A}^{3}\$(E+F)S^{2}+\$aS_{A}\$FS^{4} -\$a\$(E+F)FS^{4})\) .
