X3141
X106 -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 { 125, 2776 }: Central Line L(125, 2776)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) 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_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_2776 = ISOGONAL CONJUGATE OF X(2692) (0.8841, 0.6735) X_3141 = -CEVA CONJUGATE OF X(523) (0.2422, -0.5725)
A
B
C
X106
X523
X2
X3
X125
X2776
X3141
(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 { 125, 2776 }: Central Line L(125, 2776)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) 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_125 = CENTER OF JERABEK HYPERBOLA (0.6400, 0.0000) X_2776 = ISOGONAL CONJUGATE OF X(2692) (0.8917, 0.6200) X_3141 = -CEVA CONJUGATE OF X(523) (0.4109, -0.3055)
A
B
C
X106
X523
X2
X3
X125
X2776
X3141
(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 { 125, 2776 }: Central Line L(125, 2776)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) 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_125 = CENTER OF JERABEK HYPERBOLA (0.6180, 0.0000) X_2776 = ISOGONAL CONJUGATE OF X(2692) (0.9234, 0.5899) X_3141 = -CEVA CONJUGATE OF X(523) (0.4562, -0.1803)
A
B
C
X106
X523
X2
X3
X125
X2776
X3141
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 { 125, 2776 }: Central Line L(125, 2776)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_106 = Λ(INCENTER, CENTROID) (1.0294, -0.0621) 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_125 = CENTER OF JERABEK HYPERBOLA (0.5678, -0.0079) X_2776 = ISOGONAL CONJUGATE OF X(2692) (0.9106, 0.6377) X_3141 = -CEVA CONJUGATE OF X(523) (0.5244, -0.0725)
A
B
C
X106
X523
X2
X3
X125
X2776
X3141
Kimberling (6, 9, 13) | AB = 1
X3141 (X106 -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
\(x : y : z\) =
X106 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): \(4.500434644925\)
▸ Incident Central Lines2 lines
▸ Functional Relations1 relation
Notes
As a point on the Euler line, X3141 \((\$aS_{A}^{4}\$ -\$aS_{A}^{2}\$[(E+F)^{2}-2S^{2}] +2\$aS_{B}S_{C}\$S^{2} +2\$aS_{A}\$FS^{2}-2\$a\$(E+F)FS^{2}, 3\$aS_{A}^{2}\$S^{2} -\$a\$[(E+F)^{2}-2S^{2}]S^{2})\) .
