X598
ISOGONAL CONJUGATE OF X574
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) { 2, 187 }: Central Line L(2, 187) { 30, 262 }: Central Line L(30, 262) { 98, 381 }: Central Line L(98, 381) { 4, 575 }: Central Line L(4, 575) { 76, 524 }: Central Line L(76, 524) { 6, 671 }: Central Line L(6, 671)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_574 = HARMONIC OF X(187) (0.5539, 0.1910) X_2 = CENTROID (0.5374, 0.2000) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_262 = ISOGONAL CONJUGATE OF X(182) (0.5893, 0.2856) X_98 = TARRY POINT (0.8891, -0.2281) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_4 = ORTHOCENTER (0.6122, 0.3958) X_575 = ORTHOGONAL OF X(187) (0.5750, 0.2259) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_524 = ISOGONAL CONJUGATE OF X(111) (0.1376, -0.2297) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_671 = TRILINEAR POLE OF LINE X(2)X(523) (0.9639, -0.0323) X_598 = ISOGONAL CONJUGATE OF X(574) (0.5865, 0.2783)
A
B
C
X574
X2
X187
X30
X262
X98
X381
X4
X575
X76
X524
X6
X671
X598
(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) { 2, 187 }: Central Line L(2, 187) { 30, 262 }: Central Line L(30, 262) { 98, 381 }: Central Line L(98, 381) { 4, 575 }: Central Line L(4, 575) { 76, 524 }: Central Line L(76, 524) { 6, 671 }: Central Line L(6, 671)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_574 = HARMONIC OF X(187) (0.5828, 0.1419) X_2 = CENTROID (0.5467, 0.1600) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (0.9534, 0.7772) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_262 = ISOGONAL CONJUGATE OF X(182) (0.6225, 0.2900) X_98 = TARRY POINT (0.8489, -0.3581) X_381 = MIDPOINT OF X(2) AND X(4) (0.5933, 0.3200) X_4 = ORTHOCENTER (0.6400, 0.4800) X_575 = ORTHOGONAL OF X(187) (0.6050, 0.1800) X_76 = 3rd BROCARD POINT (0.4124, 0.0899) X_524 = ISOGONAL CONJUGATE OF X(111) (0.1101, -0.2142) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_671 = TRILINEAR POLE OF LINE X(2)X(523) (0.9422, -0.0381) X_598 = ISOGONAL CONJUGATE OF X(574) (0.6147, 0.2633)
A
B
C
X574
X2
X187
X30
X262
X98
X381
X4
X575
X76
X524
X6
X671
X598
(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) { 2, 187 }: Central Line L(2, 187) { 30, 262 }: Central Line L(30, 262) { 98, 381 }: Central Line L(98, 381) { 4, 575 }: Central Line L(4, 575) { 76, 524 }: Central Line L(76, 524) { 6, 671 }: Central Line L(6, 671)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_574 = HARMONIC OF X(187) (0.5691, 0.1422) X_2 = CENTROID (0.5393, 0.1620) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (0.9045, 0.8325) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_262 = ISOGONAL CONJUGATE OF X(182) (0.6030, 0.2930) X_98 = TARRY POINT (0.8090, -0.3931) X_381 = MIDPOINT OF X(2) AND X(4) (0.5787, 0.3239) X_4 = ORTHOCENTER (0.6180, 0.4859) X_575 = ORTHOGONAL OF X(187) (0.5885, 0.1822) X_76 = 3rd BROCARD POINT (0.4271, 0.0928) X_524 = ISOGONAL CONJUGATE OF X(111) (0.1386, -0.2504) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_671 = TRILINEAR POLE OF LINE X(2)X(523) (0.8877, -0.0692) X_598 = ISOGONAL CONJUGATE OF X(574) (0.5968, 0.2675)
A
B
C
X574
X2
X187
X30
X262
X98
X381
X4
X575
X76
X524
X6
X671
X598
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) { 2, 187 }: Central Line L(2, 187) { 30, 262 }: Central Line L(30, 262) { 98, 381 }: Central Line L(98, 381) { 4, 575 }: Central Line L(4, 575) { 76, 524 }: Central Line L(76, 524) { 6, 671 }: Central Line L(6, 671)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_574 = HARMONIC OF X(187) (0.6184, 0.0566) X_2 = CENTROID (0.5444, 0.0934) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (0.7344, 0.3809) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_262 = ISOGONAL CONJUGATE OF X(182) (0.6389, 0.3578) X_98 = TARRY POINT (0.7487, -0.7881) X_381 = MIDPOINT OF X(2) AND X(4) (0.5888, 0.4614) X_4 = ORTHOCENTER (0.6331, 0.8294) X_575 = ORTHOGONAL OF X(187) (0.6180, 0.0554) X_76 = 3rd BROCARD POINT (0.3495, 0.0360) X_524 = ISOGONAL CONJUGATE OF X(111) (-0.0086, -0.2597) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_671 = TRILINEAR POLE OF LINE X(2)X(523) (0.8374, -0.0520) X_598 = ISOGONAL CONJUGATE OF X(574) (0.6210, 0.2094)
A
B
C
X574
X2
X187
X30
X262
X98
X381
X4
X575
X76
X524
X6
X671
X598
Kimberling (6, 9, 13) | AB = 1
X598 (ISOGONAL CONJUGATE OF X574 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(bc/(a^{2} - 2b^{2} - 2c^{2}) : ca/(b^{2} - 2c^{2} - 2a^{2}) : ab/(c^{2} - 2a^{2} - 2b^{2})\)
Barycentrics: \(1/(a^{2} - 2b^{2} - 2c^{2}) : 1/(b^{2} - 2c^{2} - 2a^{2}) : 1/(c^{2} - 2a^{2} - 2b^{2})\)
Search Value (6, 9, 13): \(0.826083882460\)
▸ Incident Central Lines6 lines
{2 187 4 575 6 671 30 262 76 524 98 381
▸ Functional Relations8 relations
Notes
The Lemoine ellipse is the ellipse inscribed in triangle ABC having X2 X6 A' be where this ellipse meets sideline BC , and define B' and C' cyclically. Then triangles ABC and A'B'C' are perspective, and their perspector is X598 Bernard Gibert , 1/5/01, Hyacinthos #2334 )
