X2497
RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, 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) { 513, 676 }: Central Line L(513, 676) { 512, 2021 }: Central Line L(512, 2021)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_676 = CROSSDIFFERENCE OF X(3) AND X(101) (1.2449, 0.5665) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_2021 = RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE (0.8882, 0.7432) X_2497 = RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, GALLATLY CIRCLE} (0.6018, 0.9166)
A
B
C
X513
X676
X512
X2021
X2497
(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) { 513, 676 }: Central Line L(513, 676) { 512, 2021 }: Central Line L(512, 2021)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_676 = CROSSDIFFERENCE OF X(3) AND X(101) (1.1200, 0.3400) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) X_2021 = RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE (0.7836, 0.4861) X_2497 = RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, GALLATLY CIRCLE} (0.5186, 0.6407)
A
B
C
X513
X676
X512
X2021
X2497
(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) { 513, 676 }: Central Line L(513, 676) { 512, 2021 }: Central Line L(512, 2021)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_676 = CROSSDIFFERENCE OF X(3) AND X(101) (1.2021, 0.3440) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) X_2021 = RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE (0.7500, 0.5145) X_2497 = RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, GALLATLY CIRCLE} (0.5000, 0.6360)
A
B
C
X513
X676
X512
X2021
X2497
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) { 513, 676 }: Central Line L(513, 676) { 512, 2021 }: Central Line L(512, 2021)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_676 = CROSSDIFFERENCE OF X(3) AND X(101) (1.1010, 0.1540) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) X_2021 = RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE (0.6881, 0.2515) X_2497 = RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, GALLATLY CIRCLE} (0.4087, 0.3514)
A
B
C
X513
X676
X512
X2021
X2497
Kimberling (6, 9, 13) | AB = 1
X2497 (RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, 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 \(f(a,b,c) = a(b - c)[(b + c)(b^{2}c^{2}(b - c)^{2} + a^{4}(b^{2} + c^{2}) +a^{2}(b^{2} - bc + c^{2})^{2}) - 2a^{3}(b^{4} + b^{2}c^{2} + c^{4})]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.033618655412\)
▸ Incident Central Lines2 lines
