X2071
INVERSE-IN-CIRCUMCIRCLE OF X20
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, 403 }: Euler Line { 2, 3 }: Euler Line { 74, 323 }: Central Line L(74, 323) { 99, 1236 }: Central Line L(99, 1236) { 477, 925 }: Central Line L(477, 925) { 476, 1294 }: Central Line L(476, 1294)
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_403 = OF THE ORTHIC TRIANGLE (0.7040, 0.6358) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_323 = ISOTOMIC CONJUGATE OF X(94) (-0.8839, -1.3275) X_99 = STEINER POINT (0.1109, 0.4322) X_1236 = ISOTOMIC CONJUGATE OF X(1177) (0.1751, -0.3263) X_477 = TIXIER ANTIPODE (0.0176, -0.0644) X_925 = ANTICOMPLEMENT OF X(136) (0.0074, -0.0312) X_476 = TIXIER POINT (0.9824, 0.2685) X_1294 = COLLINGS TRANSFORM OF X(122) (0.4267, -0.4030) X_2071 = INVERSE-IN-CIRCUMCIRCLE OF X(20) (0.2043, -0.6716)
A
B
C
Ka
Kb
Kc
X2
X403
X3
X74
X323
X99
X1236
X477
X925
X476
X1294
X2071
(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, 403 }: Euler Line { 2, 3 }: Euler Line { 74, 323 }: Central Line L(74, 323) { 99, 1236 }: Central Line L(99, 1236) { 477, 925 }: Central Line L(477, 925) { 476, 1294 }: Central Line L(476, 1294)
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_403 = OF THE ORTHIC TRIANGLE (0.6400, 0.4800) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6400, -0.4800) X_323 = ISOTOMIC CONJUGATE OF X(94) (-0.2000, -0.4800) X_99 = STEINER POINT (0.1511, 0.3581) X_1236 = ISOTOMIC CONJUGATE OF X(1177) (0.2762, -0.1437) X_477 = TIXIER ANTIPODE (0.1239, -0.3295) X_925 = ANTICOMPLEMENT OF X(136) (0.3600, -0.4800) X_476 = TIXIER POINT (0.8761, 0.3295) X_1294 = COLLINGS TRANSFORM OF X(122) (0.3600, -0.4800) X_2071 = INVERSE-IN-CIRCUMCIRCLE OF X(20) (0.3600, -0.4800)
A
B
C
Ka
Kb
Kc
X2
X403
X3
X74
X323
X99
X1236
X477
X925
X476
X1294
X2071
(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, 403 }: Euler Line { 2, 3 }: Euler Line { 74, 323 }: Central Line L(74, 323) { 99, 1236 }: Central Line L(99, 1236) { 477, 925 }: Central Line L(477, 925) { 476, 1294 }: Central Line L(476, 1294)
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_403 = OF THE ORTHIC TRIANGLE (0.6180, 0.4859) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.6180, -0.4859) X_323 = ISOTOMIC CONJUGATE OF X(94) (-0.0902, -0.4859) X_99 = STEINER POINT (0.1910, 0.3931) X_1236 = ISOTOMIC CONJUGATE OF X(1177) (0.3090, -0.1501) X_477 = TIXIER ANTIPODE (0.1722, -0.3776) X_925 = ANTICOMPLEMENT OF X(136) (0.3820, -0.4859) X_476 = TIXIER POINT (0.8278, 0.3776) X_1294 = COLLINGS TRANSFORM OF X(122) (0.3820, -0.4859) X_2071 = INVERSE-IN-CIRCUMCIRCLE OF X(20) (0.3820, -0.4859)
A
B
C
Ka
Kb
Kc
X2
X403
X3
X74
X323
X99
X1236
X477
X925
X476
X1294
X2071
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, 403 }: Euler Line { 2, 3 }: Euler Line { 74, 323 }: Central Line L(74, 323) { 99, 1236 }: Central Line L(99, 1236) { 477, 925 }: Central Line L(477, 925) { 476, 1294 }: Central Line L(476, 1294)
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_403 = OF THE ORTHIC TRIANGLE (0.5841, 0.4226) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.5025, -0.8451) X_323 = ISOTOMIC CONJUGATE OF X(94) (0.3898, -0.0054) X_99 = STEINER POINT (0.2513, 0.2387) X_1236 = ISOTOMIC CONJUGATE OF X(1177) (0.3241, -0.0351) X_477 = TIXIER ANTIPODE (0.3619, -0.8282) X_925 = ANTICOMPLEMENT OF X(136) (0.7747, 0.2253) X_476 = TIXIER POINT (0.6381, 0.2788) X_1294 = COLLINGS TRANSFORM OF X(122) (0.4090, -0.8379) X_2071 = INVERSE-IN-CIRCUMCIRCLE OF X(20) (0.4650, -0.5652)
A
B
C
Ka
Kb
Kc
X2
X403
X3
X74
X323
X99
X1236
X477
X925
X476
X1294
X2071
Kimberling (6, 9, 13) | AB = 1
X2071 (INVERSE-IN-CIRCUMCIRCLE OF X20 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where
\(f(a,b,c) = (J^{2} + 1) \cos A - 2 \cos B \cos C\) , where J is as at
X1113 Search Value (6, 9, 13): \(10.061006031554\)
▸ Linear Combinations1 combination
▸ Incident Central Lines8 lines
{2 3 64 2063 74 323 99 1236 476 1294 477 925 691 1297 1290 1295
▸ Functional Relations14 relations
Notes
As a point on the Euler line, X2071 \((J^{2} + 1, -J^{2} - 3)\) .
Let P and Q be circumcircle antipodes. X2071 X4 X4
