X94
ISOGONAL CONJUGATE OF X50
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) { 49, 93 }: Central Line L(49, 93) { 23, 98 }: Central Line L(23, 98) { 4, 143 }: Central Line L(4, 143) { 2, 300 }: Central Line L(2, 300) { 275, 324 }: Central Line L(275, 324) { 96, 925 }: Central Line L(96, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_50 = -CEVA CONJUGATE OF X(184) (2.1559, 2.8368) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_93 = ISOGONAL CONJUGATE OF X(49) (0.5299, 0.2878) X_23 = FAR-OUT POINT (1.3871, 2.4232) X_98 = TARRY POINT (0.8891, -0.2281) X_4 = ORTHOCENTER (0.6122, 0.3958) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_2 = CENTROID (0.5374, 0.2000) X_300 = ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER (0.5239, 0.2121) X_275 = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT (0.6057, 0.3473) X_324 = ISOTOMIC CONJUGATE OF X(97) (0.5855, 0.3796) X_96 = ISOGONAL CONJUGATE OF X(52) (-0.0943, -0.0197) X_925 = ANTICOMPLEMENT OF X(136) (0.0074, -0.0312) X_94 = ISOGONAL CONJUGATE OF X(50) (0.9069, -0.1331)
A
B
C
X50
X49
X93
X23
X98
X4
X143
X2
X300
X275
X324
X96
X925
X94
(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) { 49, 93 }: Central Line L(49, 93) { 23, 98 }: Central Line L(23, 98) { 4, 143 }: Central Line L(4, 143) { 2, 300 }: Central Line L(2, 300) { 96, 925 }: Central Line L(96, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_50 = -CEVA CONJUGATE OF X(184) (2.2857, 3.0612) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7800, 0.0000) X_93 = ISOGONAL CONJUGATE OF X(49) (0.6400, 0.4800) X_23 = FAR-OUT POINT (0.9200, 1.4400) X_98 = TARRY POINT (0.8489, -0.3581) X_4 = ORTHOCENTER (0.6400, 0.4800) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.7120, 0.2331) X_2 = CENTROID (0.5467, 0.1600) X_300 = ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER (0.5239, 0.1906) X_96 = ISOGONAL CONJUGATE OF X(52) (1.7371, 0.1600) X_925 = ANTICOMPLEMENT OF X(136) (0.3600, -0.4800) X_94 = ISOGONAL CONJUGATE OF X(50) (0.8532, -0.2508)
A
B
C
X50
X49
X93
X23
X98
X4
X143
X2
X300
X96
X925
X94
(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) { 49, 93 }: Central Line L(49, 93) { 23, 98 }: Central Line L(23, 98) { 4, 143 }: Central Line L(4, 143) { 2, 300 }: Central Line L(2, 300) { 96, 925 }: Central Line L(96, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_50 = -CEVA CONJUGATE OF X(184) (2.6180, 4.3593) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7361, 0.0000) X_93 = ISOGONAL CONJUGATE OF X(49) (0.6180, 0.4859) X_23 = FAR-OUT POINT (0.8541, 1.4576) X_98 = TARRY POINT (0.8090, -0.3931) X_4 = ORTHOCENTER (0.6180, 0.4859) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6491, 0.3580) X_2 = CENTROID (0.5393, 0.1620) X_300 = ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER (0.5208, 0.1942) X_96 = ISOGONAL CONJUGATE OF X(52) (1.9514, 0.1620) X_925 = ANTICOMPLEMENT OF X(136) (0.3820, -0.4859) X_94 = ISOGONAL CONJUGATE OF X(50) (0.8111, -0.3087)
A
B
C
X50
X49
X93
X23
X98
X4
X143
X2
X300
X96
X925
X94
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) { 49, 93 }: Central Line L(49, 93) { 23, 98 }: Central Line L(23, 98) { 4, 143 }: Central Line L(4, 143) { 2, 300 }: Central Line L(2, 300) { 275, 324 }: Central Line L(275, 324) { 96, 925 }: Central Line L(96, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_50 = -CEVA CONJUGATE OF X(184) (0.5919, -0.0176) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.4207, 0.3163) X_93 = ISOGONAL CONJUGATE OF X(49) (-1.9471, 0.4523) X_23 = FAR-OUT POINT (0.6051, 0.5970) X_98 = TARRY POINT (0.7487, -0.7881) X_4 = ORTHOCENTER (0.6331, 0.8294) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6356, 0.2874) X_2 = CENTROID (0.5444, 0.0934) X_300 = ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER (0.6165, 0.2600) X_275 = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT (1.6919, 0.0632) X_324 = ISOTOMIC CONJUGATE OF X(97) (0.7058, 0.2879) X_96 = ISOGONAL CONJUGATE OF X(52) (1.1396, 0.0192) X_925 = ANTICOMPLEMENT OF X(136) (0.7747, 0.2253) X_94 = ISOGONAL CONJUGATE OF X(50) (0.6355, 0.3039)
A
B
C
X50
X49
X93
X23
X98
X4
X143
X2
X300
X275
X324
X96
X925
X94
Kimberling (6, 9, 13) | AB = 1
X94 (ISOGONAL CONJUGATE OF X50 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\csc 3A : \csc 3B : \csc 3C\)
Barycentrics: \(\sin A \csc 3A : \sin B \csc 3B : \sin C \csc 3C\)
Search Value (6, 9, 13): \(-0.265365586504\)
▸ Alternate Coordinates1 form
X94
▸ Incident Central Lines6 lines
{2 300 4 143 23 98 49 93 96 925 275 324
▸ Functional Relations14 relations
X94 X50
X94 isotomic conjugate of X323
X94 anticomplement of X34834
X94 cevapoint of X49 X50
X94 i -cross conjugate of Xj for these (i,j): (30 264 50 93 265 328
X94 X300 Hirst inverse of X301
X94 trilinear pole of PU(5) (line X5 X523
X94 trilinear polar of X186
X94 X48 isoconjugate (polar conjugate) of X186
X94 X476 X850
X94
X94 X74 X107 X4
X94 X476 X110
X94 A , B , C ,X6 X13 X16 X13 A , B , C ,X6 X14 X15 X14
Notes
Let A1 B1 C1 and A2 B2 C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the trilinear pole of line A1 A2 , and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X94 A1 'B1 'C1 ' and A2 'B2 'C2 ' be the 1st and 2nd Ehrmann inscribed triangles. Then X94 A2 ' , BB1’B2 ' , CC1’C2 ' . (Randy Hutson , June 27, 2018)
Let A'B'C' be the medial or orthic triangle of ABC and A″ , B″ , C″ the centers of circles {A' , X3 X4 B' , X3 X4 C' , X3 X4 A″ , B″ , C″ are collinear on the trilinear polar of X94 César Lozada , March 9, 2021).
