X2506
RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, 2nd LEMOINE 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) { 2489, 2519 }: Central Line L(2489, 2519) { 6, 512 }: Central Line L(6, 512) { 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_2489 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE} (0.1726, 0.9764) X_2519 = RADICAL CENTER OF {NINE-POINT, 2nd LEMOINE, BROCARD} CIRCLES (0.6756, -0.2289) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) 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_2506 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE} (0.4241, 0.3737)
A
B
C
X2489
X2519
X6
X512
X525
X2485
X2506
(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) { 2489, 2519 }: Van Aubel Line { 6, 512 }: Central Line L(6, 512) { 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_2489 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE} (0.6400, 0.4800) X_2519 = RADICAL CENTER OF {NINE-POINT, 2nd LEMOINE, BROCARD} CIRCLES (0.6400, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) 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_2506 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE} (0.6400, 0.2400)
A
B
C
X2489
X2519
X6
X512
X525
X2485
X2506
(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) { 2489, 2519 }: Van Aubel Line { 6, 512 }: Central Line L(6, 512) { 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_2489 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE} (0.6180, 0.4859) X_2519 = RADICAL CENTER OF {NINE-POINT, 2nd LEMOINE, BROCARD} CIRCLES (0.6180, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) 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_2506 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE} (0.6180, 0.2429)
A
B
C
X2489
X2519
X6
X512
X525
X2485
X2506
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) { 2489, 2519 }: Central Line L(2489, 2519) { 6, 512 }: Central Line L(6, 512) { 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_2489 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE} (0.8221, 0.1876) X_2519 = RADICAL CENTER OF {NINE-POINT, 2nd LEMOINE, BROCARD} CIRCLES (0.6835, 0.0752) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) 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_2506 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE} (0.7528, 0.1314)
A
B
C
X2489
X2519
X6
X512
X525
X2485
X2506
Kimberling (6, 9, 13) | AB = 1
X2506 (RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, 2nd LEMOINE 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 \(f(a,b,c) = a(b^{2} - c^{2})[a^{6} + (b^{2} + c^{2})(a^{4} - b^{4} + 4b^{2}c^{2} - c^{4}) - a^{2}(b^{4} + 4b^{2}c^{2} + c^{4})]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.592698578987\)
▸ Incident Central Lines3 lines
▸ Functional Relations1 relation
