X2491
RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-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) { 230, 231 }: Central Line L(230, 231) { 39, 690 }: Central Line L(39, 690) { 351, 865 }: Central Line L(351, 865) { 669, 688 }: Central Line L(669, 688) { 512, 2021 }: Central Line L(512, 2021) { 804, 2023 }: Central Line L(804, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_231 = -CEVA CONJUGATE OF X(128) (1.0375, 0.6458) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_690 = CROSSDIFFERENCE OF LINE X(6) AND X(110) (0.0840, -0.1726) X_351 = CENTER OF THE PARRY CIRCLE (1.3537, 1.1619) X_865 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512) (-0.1837, -1.6869) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-4.3514, 4.6163) 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_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_2491 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, GALLATLY CIRCLE} (1.0687, 0.6339)
A
B
C
X230
X231
X39
X690
X351
X865
X669
X688
X512
X2021
X804
X2023
X2491
(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) { 230, 231 }: Central Line L(230, 231) { 39, 690 }: Central Line L(39, 690) { 351, 865 }: Central Line L(351, 865) { 669, 688 }: Central Line L(669, 688) { 512, 2021 }: Central Line L(512, 2021) { 804, 2023 }: Central Line L(804, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_230 = -CEVA CONJUGATE OF X(114) (0.8489, 0.4191) X_231 = -CEVA CONJUGATE OF X(128) (0.9181, 0.3989) X_39 = BROCARD MIDPOINT (0.6138, 0.1951) X_690 = CROSSDIFFERENCE OF LINE X(6) AND X(110) (0.0500, -0.1297) X_351 = CENTER OF THE PARRY CIRCLE (1.0952, 0.6944) X_865 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512) (0.1047, -1.3552) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-1.2857, 2.0833) 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_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_2491 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, GALLATLY CIRCLE} (0.9482, 0.3901)
A
B
C
X230
X231
X39
X690
X351
X865
X669
X688
X512
X2021
X804
X2023
X2491
(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) { 230, 231 }: Central Line L(230, 231) { 39, 690 }: Central Line L(39, 690) { 351, 865 }: Central Line L(351, 865) { 669, 688 }: Central Line L(669, 688) { 512, 2021 }: Central Line L(512, 2021) { 804, 2023 }: Central Line L(804, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_230 = -CEVA CONJUGATE OF X(114) (0.8090, 0.4395) X_231 = -CEVA CONJUGATE OF X(128) (0.8795, 0.4223) X_39 = BROCARD MIDPOINT (0.5955, 0.1965) X_690 = CROSSDIFFERENCE OF LINE X(6) AND X(110) (0.0222, -0.0893) X_351 = CENTER OF THE PARRY CIRCLE (1.2060, 0.6861) X_865 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512) (0.3820, -0.4859) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-1.6180, 2.0582) 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_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_2491 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, GALLATLY CIRCLE} (1.0000, 0.3931)
A
B
C
X230
X231
X39
X690
X351
X865
X669
X688
X512
X2021
X804
X2023
X2491
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) { 230, 231 }: Central Line L(230, 231) { 39, 690 }: Central Line L(39, 690) { 351, 865 }: Central Line L(351, 865) { 669, 688 }: Central Line L(669, 688) { 512, 2021 }: Central Line L(512, 2021) { 804, 2023 }: Central Line L(804, 2023)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_230 = -CEVA CONJUGATE OF X(114) (0.7126, 0.2008) X_231 = -CEVA CONJUGATE OF X(128) (1.0516, 0.1599) X_39 = BROCARD MIDPOINT (0.6418, 0.1220) X_690 = CROSSDIFFERENCE OF LINE X(6) AND X(110) (-0.0947, -0.0548) X_351 = CENTER OF THE PARRY CIRCLE (0.9667, 0.2979) X_865 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512) (0.4786, -0.4519) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-0.1118, 0.6834) 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_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_2491 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, GALLATLY CIRCLE} (0.8896, 0.1795)
A
B
C
X230
X231
X39
X690
X351
X865
X669
X688
X512
X2021
X804
X2023
X2491
Kimberling (6, 9, 13) | AB = 1
X2491 (RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, GALLATLY CIRCLE} ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a^{3}(b^{2} - c^{2})(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2}) : :\)
Search Value (6, 9, 13): \(-0.983274281501\)
▸ Incident Central Lines8 lines
{39 690 230 231 351 865 512 2021 669 688 684 3569 804 2023 882 2422
▸ Functional Relations8 relations
X2491 i -Ceva conjugate of Xj for these (i,j): (882 688 2395 512 2396 511
X2491 crosspoint of Xi and Xj for these (i,j): (2 805 112 1976 511 2396 512 2395 669 881
X2491 crosssum of Xi and Xj for these (i,j): (6 804 98 2422 99 2421 325 525 645 874 670 880
X2491 X2679
X2491 trilinear poles of lines passing through X2679
X2491 X2 X2679
X2491 X511 X512
X2491 X511 X512
Notes
Let A'B'C' and A″B″C″ be the circumcevian triangles of X511 X512 A* be the barycentric product A' *A″ , and define B* and C* cyclically. The lines AA* , BB* , CC* concur in X2491 Randy Hutson , October 8, 2019)
