X2965
BARYCENTRIC PRODUCT X61 *X62
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) { 3, 6 }: Brocard Axis { 53, 112 }: Central Line L(53, 112) { 524, 1238 }: Central Line L(524, 1238) { 230, 1627 }: Central Line L(230, 1627) { 5, 2963 }: Central Line L(5, 2963)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_524 = ISOGONAL CONJUGATE OF X(111) (0.1376, -0.2297) X_1238 = ISOTOMIC CONJUGATE OF X(1179) (0.3603, 0.0438) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_1627 = TCC-PERSPECTOR OF X(251) (0.6795, 0.3846) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2963 = BARYCENTRIC PRODUCT X(17)*X(18) (0.4808, 0.1641) X_2965 = BARYCENTRIC PRODUCT X(61)*X(62) (0.6592, 0.3650)
A
B
C
X3
X6
X53
X112
X524
X1238
X230
X1627
X5
X2963
X2965
(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) { 3, 6 }: Brocard Axis { 524, 1238 }: Central Line L(524, 1238) { 230, 1627 }: Central Line L(230, 1627) { 5, 2963 }: Central Line L(5, 2963)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_524 = ISOGONAL CONJUGATE OF X(111) (0.1101, -0.2142) X_1238 = ISOTOMIC CONJUGATE OF X(1179) (0.2979, 0.0000) X_230 = -CEVA CONJUGATE OF X(114) (0.8489, 0.4191) X_1627 = TCC-PERSPECTOR OF X(251) (0.7211, 0.3558) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_2963 = BARYCENTRIC PRODUCT X(17)*X(18) (0.3993, 0.1024) X_2965 = BARYCENTRIC PRODUCT X(61)*X(62) (0.7021, 0.3464)
A
B
C
X3
X6
X524
X1238
X230
X1627
X5
X2963
X2965
(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) { 3, 6 }: Brocard Axis { 524, 1238 }: Central Line L(524, 1238) { 230, 1627 }: Central Line L(230, 1627) { 5, 2963 }: Central Line L(5, 2963)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_524 = ISOGONAL CONJUGATE OF X(111) (0.1386, -0.2504) X_1238 = ISOTOMIC CONJUGATE OF X(1179) (0.3277, 0.0000) X_230 = -CEVA CONJUGATE OF X(114) (0.8090, 0.4395) X_1627 = TCC-PERSPECTOR OF X(251) (0.6893, 0.3651) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_2963 = BARYCENTRIC PRODUCT X(17)*X(18) (0.4238, 0.1097) X_2965 = BARYCENTRIC PRODUCT X(61)*X(62) (0.6723, 0.3545)
A
B
C
X3
X6
X524
X1238
X230
X1627
X5
X2963
X2965
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) { 3, 6 }: Brocard Axis { 53, 112 }: Central Line L(53, 112) { 524, 1238 }: Central Line L(524, 1238) { 230, 1627 }: Central Line L(230, 1627) { 5, 2963 }: Central Line L(5, 2963)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6533, 0.2777) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.6732, 0.2689) X_524 = ISOGONAL CONJUGATE OF X(111) (-0.0086, -0.2597) X_1238 = ISOTOMIC CONJUGATE OF X(1179) (0.2791, -0.0025) X_230 = -CEVA CONJUGATE OF X(114) (0.7126, 0.2008) X_1627 = TCC-PERSPECTOR OF X(251) (0.6986, 0.2405) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_2963 = BARYCENTRIC PRODUCT X(17)*X(18) (1.4997, 0.1536) X_2965 = BARYCENTRIC PRODUCT X(61)*X(62) (0.6914, 0.2608)
A
B
C
X3
X6
X53
X112
X524
X1238
X230
X1627
X5
X2963
X2965
Kimberling (6, 9, 13) | AB = 1
X2965 (BARYCENTRIC PRODUCT X61 *X62 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sin A \sin(A + \pi /6) \sin(A - \pi /6)\)
Search Value (6, 9, 13): \(-0.260809398448\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines5 lines
{3 6 5 2963 53 112 230 1627 524 1238
▸ Functional Relations3 relations
Notes
Let OA OB OC be the Kosnita triangle . Let A″ be the barycentric product of the circumcircle intercepts of line OB OC . Define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X2965 Randy Hutson , July 31 2018)
