X311
ISOTOMIC CONJUGATE OF X54
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC < K_a K_b K_c >: Anticomplementary Triangle △K_a K_b K_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 570 }: Central Line L(2, 570) { 4, 69 }: Central Line L(4, 69) { 22, 157 }: Central Line L(22, 157) { 95, 99 }: Central Line L(95, 99) { 290, 308 }: Central Line L(290, 308)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) K_a = Anticomplementary Vertex K_a (1.6122, 0.5999) K_b = Anticomplementary Vertex K_b (-0.3878, 0.5999) K_c = Anticomplementary Vertex K_c (0.3878, -0.5999) X_2 = CENTROID (0.5374, 0.2000) X_570 = ORTHOGONAL OF X(52) (0.5652, 0.2098) X_4 = ORTHOCENTER (0.6122, 0.3958) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_22 = EXETER POINT (0.4143, -0.1221) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5266, 0.3812) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5172, 0.1562) X_99 = STEINER POINT (0.1109, 0.4322) X_290 = ISOGONAL CONJUGATE OF X(237) (0.7829, -0.1380) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4915, 0.1700) X_311 = ISOTOMIC CONJUGATE OF X(54) (0.4818, 0.1803)
A
B
C
Ka
Kb
Kc
X2
X570
X4
X69
X22
X157
X95
X99
X290
X308
X311
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC < K_a K_b K_c >: Anticomplementary Triangle △K_a K_b K_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 570 }: Central Line L(2, 570) { 4, 69 }: Central Line L(4, 69) { 22, 157 }: Central Line L(22, 157) { 95, 99 }: Central Line L(95, 99) { 290, 308 }: Central Line L(290, 308)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) K_a = Anticomplementary Vertex K_a (1.6400, 0.4800) K_b = Anticomplementary Vertex K_b (-0.3600, 0.4800) K_c = Anticomplementary Vertex K_c (0.3600, -0.4800) X_2 = CENTROID (0.5467, 0.1600) X_570 = ORTHOGONAL OF X(52) (0.5958, 0.1643) X_4 = ORTHOCENTER (0.6400, 0.4800) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_22 = EXETER POINT (0.3600, -0.4800) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5000, 0.5208) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5280, 0.0960) X_99 = STEINER POINT (0.1511, 0.3581) X_290 = ISOGONAL CONJUGATE OF X(237) (0.6400, -0.1437) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4655, 0.1249) X_311 = ISOTOMIC CONJUGATE OF X(54) (0.4483, 0.1514)
A
B
C
Ka
Kb
Kc
X2
X570
X4
X69
X22
X157
X95
X99
X290
X308
X311
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC < K_a K_b K_c >: Anticomplementary Triangle △K_a K_b K_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 570 }: Central Line L(2, 570) { 4, 69 }: Central Line L(4, 69) { 22, 157 }: Central Line L(22, 157) { 95, 99 }: Central Line L(95, 99) { 290, 308 }: Central Line L(290, 308)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) K_a = Anticomplementary Vertex K_a (1.6180, 0.4859) K_b = Anticomplementary Vertex K_b (-0.3820, 0.4859) K_c = Anticomplementary Vertex K_c (0.3820, -0.4859) X_2 = CENTROID (0.5393, 0.1620) X_570 = ORTHOGONAL OF X(52) (0.5802, 0.1650) X_4 = ORTHOCENTER (0.6180, 0.4859) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_22 = EXETER POINT (0.3820, -0.4859) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5000, 0.5145) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5236, 0.0972) X_99 = STEINER POINT (0.1910, 0.3931) X_290 = ISOGONAL CONJUGATE OF X(237) (0.6180, -0.1501) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4721, 0.1282) X_311 = ISOTOMIC CONJUGATE OF X(54) (0.4577, 0.1558)
A
B
C
Ka
Kb
Kc
X2
X570
X4
X69
X22
X157
X95
X99
X290
X308
X311
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC < K_a K_b K_c >: Anticomplementary Triangle △K_a K_b K_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 570 }: Central Line L(2, 570) { 4, 69 }: Central Line L(4, 69) { 22, 157 }: Central Line L(22, 157) { 95, 99 }: Central Line L(95, 99) { 290, 308 }: Central Line L(290, 308)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) K_a = Anticomplementary Vertex K_a (1.6331, 0.2801) K_b = Anticomplementary Vertex K_b (-0.3669, 0.2801) K_c = Anticomplementary Vertex K_c (0.3669, -0.2801) X_2 = CENTROID (0.5444, 0.0934) X_570 = ORTHOGONAL OF X(52) (0.6067, 0.0238) X_4 = ORTHOCENTER (0.6331, 0.8294) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_22 = EXETER POINT (0.8328, 2.4854) X_157 = -OF-TANGENTIAL-TRIANGLE (0.4713, 0.5134) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (12.6715, -0.2255) X_99 = STEINER POINT (0.2513, 0.2387) X_290 = ISOGONAL CONJUGATE OF X(237) (0.4119, -0.0928) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4157, 0.0638) X_311 = ISOTOMIC CONJUGATE OF X(54) (0.4197, 0.2324)
A
B
C
Ka
Kb
Kc
X2
X570
X4
X69
X22
X157
X95
X99
X290
X308
X311
Kimberling (6, 9, 13) | AB = 1
X311 (ISOTOMIC CONJUGATE OF X54 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\csc^{2} A \cos(B - C) : :\)
Barycentrics: \(\csc A \cos(B - C) : \csc B \cos(C - A) : \csc C \cos(A - B)\)
Search Value (6, 9, 13): \(2.175140143881\)
▸ Incident Central Lines10 lines
{2 570 4 69 22 157 53 324 95 99 141 338 290 308 300 303 301 302 305 1007
▸ Functional Relations9 relations
Notes
Let OA OB OC be the Kosnita triangle . Let A' be the trilinear pole of line OB OC , and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X1994 A″ be the trilinear pole of line B'C' , and define B″and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X311 Randy Hutson , November 30, 2018)
