X2513
RADICAL CENTER OF {CIRCUMCIRCLE, GALLATLY CIRCLE, (X4 ,2R)}
The circle (
X4 ,2R) is identified at
X2474 .
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) { 325, 523 }: Central Line L(325, 523) { 647, 688 }: Central Line L(647, 688) { 512, 2021 }: Central Line L(512, 2021) { 525, 2531 }: Central Line L(525, 2531)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_325 = -HIRST INVERSE OF X(69) (-0.3370, -0.7398) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (4.2063, -0.5653) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0103, -0.1475) 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_525 = ISOGONAL CONJUGATE OF X(112) (-0.0468, 0.2556) X_2531 = RADICAL CENTER OF {BROCARD, (X(4),2R), 2nd BROCARD} CIRCLES (8.2533, -4.4169) X_2513 = RADICAL CENTER OF {CIRCUMCIRCLE, GALLATLY CIRCLE, (X(4),2R)} (9.6264, -4.5477)
A
B
C
X325
X523
X647
X688
X512
X2021
X525
X2531
X2513
(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) { 325, 523 }: Central Line L(325, 523) { 647, 688 }: Central Line L(647, 688) { 512, 2021 }: Central Line L(512, 2021) { 525, 2531 }: Central Line L(525, 2531)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_325 = -HIRST INVERSE OF X(69) (-0.0578, -0.3581) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0987, -0.0010) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (2.2857, 0.0000) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0049, -0.1873) 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_525 = ISOGONAL CONJUGATE OF X(112) (-0.0283, 0.1600) X_2531 = RADICAL CENTER OF {BROCARD, (X(4),2R), 2nd BROCARD} CIRCLES (3.8508, -1.6932) X_2513 = RADICAL CENTER OF {CIRCUMCIRCLE, GALLATLY CIRCLE, (X(4),2R)} (4.5196, -1.6932)
A
B
C
X325
X523
X647
X688
X512
X2021
X525
X2531
X2513
(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) { 325, 523 }: Central Line L(325, 523) { 647, 688 }: Central Line L(647, 688) { 512, 2021 }: Central Line L(512, 2021) { 525, 2531 }: Central Line L(525, 2531)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_325 = -HIRST INVERSE OF X(69) (0.0000, -0.3931) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0981, 0.0262) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (2.6180, 0.0000) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.0245, -0.1466) 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_525 = ISOGONAL CONJUGATE OF X(112) (-0.0357, 0.1620) X_2531 = RADICAL CENTER OF {BROCARD, (X(4),2R), 2nd BROCARD} CIRCLES (4.4271, -1.6651) X_2513 = RADICAL CENTER OF {CIRCUMCIRCLE, GALLATLY CIRCLE, (X(4),2R)} (5.2361, -1.6651)
A
B
C
X325
X523
X647
X688
X512
X2021
X525
X2531
X2513
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) { 325, 523 }: Central Line L(325, 523) { 647, 688 }: Central Line L(647, 688) { 512, 2021 }: Central Line L(512, 2021) { 525, 2531 }: Central Line L(525, 2531)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_325 = -HIRST INVERSE OF X(69) (0.2079, -0.1215) X_523 = ISOGONAL CONJUGATE OF X(110) (1.1957, 0.0148) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (1.5059, 0.1051) X_688 = CROSSDIFFERENCE OF X(6) AND X(76) (1.1304, -0.2016) 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_525 = ISOGONAL CONJUGATE OF X(112) (-0.1112, 0.0694) X_2531 = RADICAL CENTER OF {BROCARD, (X(4),2R), 2nd BROCARD} CIRCLES (2.1366, -0.4123) X_2513 = RADICAL CENTER OF {CIRCUMCIRCLE, GALLATLY CIRCLE, (X(4),2R)} (2.5072, -0.3988)
A
B
C
X325
X523
X647
X688
X512
X2021
X525
X2531
X2513
Kimberling (6, 9, 13) | AB = 1
X2513 (RADICAL CENTER OF {CIRCUMCIRCLE, GALLATLY CIRCLE, (X4 ,2R)} ) 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 \(f(a,b,c) = a(b^{2} - c^{2})[a^{2}(2b^{4} + 3b^{2}c^{2} + 2c^{4}) + b^{2}c^{2}(b^{2} + c^{2})]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-7.768231947533\)
▸ Incident Central Lines4 lines
▸ Functional Relations1 relation
