X2509
RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, APOLLONIUS CIRCLE}
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) { 661, 2484 }: Central Line L(661, 2484) { 6, 521 }: Central Line L(6, 521) { 44, 513 }: Central Line L(44, 513) { 905, 918 }: Central Line L(905, 918) { 525, 2485 }: Central Line L(525, 2485)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_2484 = RADICAL CENTER OF {CIRCUMCIRCLE, 2nd LEMOINE CIRCLE, BEVAN CIRCLE} (-35.7273, 22.7135) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_521 = ISOGONAL CONJUGATE OF X(108) (1.1224, 0.1534) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_905 = POINT ACHERNAR (5.6190, -0.6999) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.0192, 0.5351) X_525 = ISOGONAL CONJUGATE OF X(112) (-0.0468, 0.2556) X_2485 = RADICAL CENTER OF {CIRCUMCIRCLE, 1ST LEMOINE CIRCLE, NINE-POINT CIRCLE} (2.1894, 0.2056) X_2509 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, APOLLONIUS CIRCLE} (6.3506, -0.1909)
A
B
C
X661
X2484
X6
X521
X44
X513
X905
X918
X525
X2485
X2509
(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) { 661, 2484 }: Central Line L(661, 2484) { 6, 521 }: Central Line L(6, 521) { 44, 513 }: Central Line L(44, 513) { 905, 918 }: Central Line L(905, 918) { 525, 2485 }: Central Line L(525, 2485)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (20.8000, -8.4000) X_2484 = RADICAL CENTER OF {CIRCUMCIRCLE, 2nd LEMOINE CIRCLE, BEVAN CIRCLE} (-13.7600, 8.8800) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_521 = ISOGONAL CONJUGATE OF X(108) (1.1217, 0.1600) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_905 = POINT ACHERNAR (3.0000, 0.0000) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.0687, 0.4010) X_525 = ISOGONAL CONJUGATE OF X(112) (-0.0283, 0.1600) X_2485 = RADICAL CENTER OF {CIRCUMCIRCLE, 1ST LEMOINE CIRCLE, NINE-POINT CIRCLE} (1.4629, 0.2400) X_2509 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, APOLLONIUS CIRCLE} (3.5200, 0.2400)
A
B
C
X661
X2484
X6
X521
X44
X513
X905
X918
X525
X2485
X2509
(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) { 661, 2484 }: Central Line L(661, 2484) { 6, 521 }: Central Line L(6, 521) { 44, 513 }: Central Line L(44, 513) { 905, 918 }: Central Line L(905, 918) { 525, 2485 }: Central Line L(525, 2485)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (24.7569, -8.3524) X_2484 = RADICAL CENTER OF {CIRCUMCIRCLE, 2nd LEMOINE CIRCLE, BEVAN CIRCLE} (-16.5727, 8.8383) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_521 = ISOGONAL CONJUGATE OF X(108) (1.1143, 0.1620) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_905 = POINT ACHERNAR (3.4741, 0.0000) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.0745, 0.3723) X_525 = ISOGONAL CONJUGATE OF X(112) (-0.0357, 0.1620) X_2485 = RADICAL CENTER OF {CIRCUMCIRCLE, 1ST LEMOINE CIRCLE, NINE-POINT CIRCLE} (1.6180, 0.2429) X_2509 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, APOLLONIUS CIRCLE} (4.0921, 0.2429)
A
B
C
X661
X2484
X6
X521
X44
X513
X905
X918
X525
X2485
X2509
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) { 661, 2484 }: Central Line L(661, 2484) { 6, 521 }: Central Line L(6, 521) { 44, 513 }: Central Line L(44, 513) { 905, 918 }: Central Line L(905, 918) { 525, 2485 }: Central Line L(525, 2485)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (9.8407, -1.9504) X_2484 = RADICAL CENTER OF {CIRCUMCIRCLE, 2nd LEMOINE CIRCLE, BEVAN CIRCLE} (-5.1673, 2.3286) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_521 = ISOGONAL CONJUGATE OF X(108) (-0.1116, 0.0841) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_905 = POINT ACHERNAR (1.9337, 0.1207) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.1912, 0.2031) X_525 = ISOGONAL CONJUGATE OF X(112) (-0.1112, 0.0694) X_2485 = RADICAL CENTER OF {CIRCUMCIRCLE, 1ST LEMOINE CIRCLE, NINE-POINT CIRCLE} (1.1640, 0.1464) X_2509 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, APOLLONIUS CIRCLE} (2.3367, 0.1891)
A
B
C
X661
X2484
X6
X521
X44
X513
X905
X918
X525
X2485
X2509
Kimberling (6, 9, 13) | AB = 1
X2509 (RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, APOLLONIUS CIRCLE} ) 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 \(f(a,b,c) = (b - c)[a^{3} - a^{2}(b + c) + a(b + c)^{2} - (b + c)(b^{2} + c^{2})]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-12.498038675985\)
▸ Incident Central Lines5 lines
▸ Functional Relations3 relations
