X3459
ISOGONAL CONJUGATE OF X195
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) { 5, 195 }: Central Line L(5, 195) { 137, 252 }: Central Line L(137, 252) { 4, 1157 }: Central Line L(4, 1157) { 3, 1263 }: Central Line L(3, 1263) { 53, 1601 }: Central Line L(53, 1601) { 288, 1487 }: Central Line L(288, 1487)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_195 = -CEVA CONJUGATE OF X(3) (0.6761, 0.3331) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_137 = -OF-ORTHIC-TRIANGLE (0.7813, 0.1289) X_252 = ISOGONAL CONJUGATE OF X(143) (0.4565, 0.1253) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1157 = INVERSE-IN-CIRCUMCIRCLE OF X(54) (1.5871, 1.5280) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1263 = ISOGONAL CONJUGATE OF X(1157) (1.0064, 0.0088) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_1601 = TCC-PERSPECTOR OF X(5) (0.4370, 0.2016) X_288 = CEVAPOINT OF X(6) AND X(54) (0.5936, 0.2442) X_1487 = NAPOLEON CEVAPOINT (0.4925, 0.1879) X_3459 = ISOGONAL CONJUGATE OF X(195) (0.3786, 0.1244)
A
B
C
X195
X5
X137
X252
X4
X1157
X3
X1263
X53
X1601
X288
X1487
X3459
(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) { 5, 195 }: Central Line L(5, 195) { 137, 252 }: Central Line L(137, 252) { 4, 1157 }: Central Line L(4, 1157) { 3, 1263 }: Central Line L(3, 1263) { 53, 1601 }: Central Line L(53, 1601) { 288, 1487 }: Central Line L(288, 1487)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_195 = -CEVA CONJUGATE OF X(3) (0.7800, 0.3200) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_137 = -OF-ORTHIC-TRIANGLE (-0.2811, -0.3630) X_252 = ISOGONAL CONJUGATE OF X(143) (0.3705, 0.0573) X_4 = ORTHOCENTER (0.6400, 0.4800) X_1157 = INVERSE-IN-CIRCUMCIRCLE OF X(54) (1.2743, 0.8850) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1263 = ISOGONAL CONJUGATE OF X(1157) (1.0272, 0.0375) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6400, 0.4800) X_1601 = TCC-PERSPECTOR OF X(5) (0.3600, 0.3013) X_288 = CEVAPOINT OF X(6) AND X(54) (0.6400, 0.2057) X_1487 = NAPOLEON CEVAPOINT (0.4225, 0.1389) X_3459 = ISOGONAL CONJUGATE OF X(195) (-0.1887, -0.0490)
A
B
C
X195
X5
X137
X252
X4
X1157
X3
X1263
X53
X1601
X288
X1487
X3459
(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) { 5, 195 }: Central Line L(5, 195) { 137, 252 }: Central Line L(137, 252) { 4, 1157 }: Central Line L(4, 1157) { 3, 1263 }: Central Line L(3, 1263) { 53, 1601 }: Central Line L(53, 1601) { 288, 1487 }: Central Line L(288, 1487)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_195 = -CEVA CONJUGATE OF X(3) (0.7361, 0.3239) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_137 = -OF-ORTHIC-TRIANGLE (0.8539, 0.1461) X_252 = ISOGONAL CONJUGATE OF X(143) (0.3991, 0.0618) X_4 = ORTHOCENTER (0.6180, 0.4859) X_1157 = INVERSE-IN-CIRCUMCIRCLE OF X(54) (1.2347, 1.0081) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_1263 = ISOGONAL CONJUGATE OF X(1157) (0.9854, -0.0182) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6180, 0.4859) X_1601 = TCC-PERSPECTOR OF X(5) (0.3820, 0.2859) X_288 = CEVAPOINT OF X(6) AND X(54) (0.6180, 0.2082) X_1487 = NAPOLEON CEVAPOINT (0.4435, 0.1478) X_3459 = ISOGONAL CONJUGATE OF X(195) (0.0636, 0.0164)
A
B
C
X195
X5
X137
X252
X4
X1157
X3
X1263
X53
X1601
X288
X1487
X3459
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) { 5, 195 }: Central Line L(5, 195) { 137, 252 }: Central Line L(137, 252) { 4, 1157 }: Central Line L(4, 1157) { 3, 1263 }: Central Line L(3, 1263) { 53, 1601 }: Central Line L(53, 1601) { 288, 1487 }: Central Line L(288, 1487)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_195 = -CEVA CONJUGATE OF X(3) (3.2601, 0.1287) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_137 = -OF-ORTHIC-TRIANGLE (0.7097, 0.5241) X_252 = ISOGONAL CONJUGATE OF X(143) (0.6562, -0.0054) X_4 = ORTHOCENTER (0.6331, 0.8294) X_1157 = INVERSE-IN-CIRCUMCIRCLE OF X(54) (0.7309, -0.2409) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_1263 = ISOGONAL CONJUGATE OF X(1157) (0.8529, 0.7708) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6533, 0.2777) X_1601 = TCC-PERSPECTOR OF X(5) (0.3443, 0.3461) X_288 = CEVAPOINT OF X(6) AND X(54) (0.5855, 0.2917) X_1487 = NAPOLEON CEVAPOINT (0.6484, 0.2784) X_3459 = ISOGONAL CONJUGATE OF X(195) (0.6841, 0.2709)
A
B
C
X195
X5
X137
X252
X4
X1157
X3
X1263
X53
X1601
X288
X1487
X3459
Kimberling (6, 9, 13) | AB = 1
X3459 (ISOGONAL CONJUGATE OF X195 ) 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 \(f(a,b,c) = y(195)z(195)\)
Search Value (6, 9, 13): \(-0.307549623304\)
X3459
▸ Incident Central Lines8 lines
{3 1263 4 1157 5 195 53 1601 137 252 288 1487 3336 3460 3462 3470
▸ Functional Relations5 relations
Notes
Let A' be the reflection of X5 BC , and define B' and C' cyclically. Let Oa be the circumcenter of A'BC , and define Ob and Oc cyclically. The lines AOa , BOb , COc concur in X3459 Randy Hutson , December 2, 2017)
