X2507
RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY 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, 2524 }: Central Line L(2489, 2524) { 512, 2024 }: Central Line L(512, 2024) { 804, 2023 }: Central Line L(804, 2023) { 1194, 1637 }: Central Line L(1194, 1637) { 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_2524 = RADICAL CENTER OF {NINE-POINT, BROCARD, 2nd BROCARD} CIRCLES (1.4600, -0.3036) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6887, 0.4136) X_804 = ODD (- 2, 2) INFINITY POINT (0.9171, 0.6474) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6642, 0.1572) X_1194 = 3rd SARAGOSSA POINT OF X(2) (0.5959, 0.2442) X_1637 = TRIPOLAR CENTROID OF X(4) (1.3086, 0.5422) 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_2507 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY CIRCLE} (0.8163, 0.3364)
A
B
C
X2489
X2524
X512
X2024
X804
X2023
X1194
X1637
X525
X2485
X2507
(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, 2524 }: Central Line L(2489, 2524) { 512, 2024 }: Central Line L(512, 2024) { 804, 2023 }: Central Line L(804, 2023) { 1194, 1637 }: Central Line L(1194, 1637) { 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_2524 = RADICAL CENTER OF {NINE-POINT, BROCARD, 2nd BROCARD} CIRCLES (0.9482, 0.0000) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6791, 0.3071) X_804 = ODD (- 2, 2) INFINITY POINT (0.9585, 0.5612) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6791, 0.1280) X_1194 = 3rd SARAGOSSA POINT OF X(2) (0.6400, 0.2088) X_1637 = TRIPOLAR CENTROID OF X(4) (1.1886, 0.3200) 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_2507 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY CIRCLE} (0.7941, 0.2400)
A
B
C
X2489
X2524
X512
X2024
X804
X2023
X1194
X1637
X525
X2485
X2507
(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, 2524 }: Central Line L(2489, 2524) { 512, 2024 }: Central Line L(512, 2024) { 804, 2023 }: Central Line L(804, 2023) { 1194, 1637 }: Central Line L(1194, 1637) { 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_2524 = RADICAL CENTER OF {NINE-POINT, BROCARD, 2nd BROCARD} CIRCLES (1.0000, 0.0000) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6545, 0.3180) X_804 = ODD (- 2, 2) INFINITY POINT (0.9914, 0.5173) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6545, 0.1215) X_1194 = 3rd SARAGOSSA POINT OF X(2) (0.6180, 0.2104) X_1637 = TRIPOLAR CENTROID OF X(4) (1.2847, 0.3239) 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_2507 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY CIRCLE} (0.8090, 0.2429)
A
B
C
X2489
X2524
X512
X2024
X804
X2023
X1194
X1637
X525
X2485
X2507
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, 2524 }: Central Line L(2489, 2524) { 512, 2024 }: Central Line L(512, 2024) { 804, 2023 }: Central Line L(804, 2023) { 1194, 1637 }: Central Line L(1194, 1637) { 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_2524 = RADICAL CENTER OF {NINE-POINT, BROCARD, 2nd BROCARD} CIRCLES (0.7647, 0.0781) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6624, 0.1797) X_804 = ODD (- 2, 2) INFINITY POINT (1.1348, 0.3793) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6664, 0.0713) X_1194 = 3rd SARAGOSSA POINT OF X(2) (0.6706, 0.1305) X_1637 = TRIPOLAR CENTROID OF X(4) (1.2096, 0.1409) 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_2507 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY CIRCLE} (0.7934, 0.1329)
A
B
C
X2489
X2524
X512
X2024
X804
X2023
X1194
X1637
X525
X2485
X2507
Kimberling (6, 9, 13) | AB = 1
X2507 (RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY 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 \(f(a,b,c) = a[(b^{4} - c^{4})a^{6} - (b^{8} - c^{8})a^{2} + 2b^{4}c^{4}(b^{2} - c^{2})]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.257171794921\)
▸ Incident Central Lines5 lines
▸ Functional Relations1 relation
