X2500
RADICAL CENTER OF {CIRCUMCIRCLE, SPIEKER CIRCLE, APOLLONIUS 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) { 512, 650 }: Central Line L(512, 650) { 522, 2490 }: Central Line L(522, 2490) { 513, 2532 }: Central Line L(513, 2532)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0318, 0.3430) X_2490 = RADICAL CENTER OF {CIRCUMCIRCLE,NINE-POINT-CIRCLE, SPIEKER CIRCLE} (-5.7092, 3.2243) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_2532 = RADICAL CENTER OF {APOLLONIUS, SPIEKER, BEVAN} CIRCLES (8.9543, -1.1690) X_2500 = RADICAL CENTER OF {CIRCUMCIRCLE, SPIEKER CIRCLE, APOLLONIUS CIRCLE} (6.5340, 0.1485)
A
B
C
X512
X650
X522
X2490
X513
X2532
X2500
(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) { 512, 650 }: Central Line L(512, 650) { 522, 2490 }: Central Line L(522, 2490) { 513, 2532 }: Central Line L(513, 2532)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0433, -0.1297) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (6.4000, -1.2000) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0226, 0.2413) X_2490 = RADICAL CENTER OF {CIRCUMCIRCLE,NINE-POINT-CIRCLE, SPIEKER CIRCLE} (-2.2400, 1.3200) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_2532 = RADICAL CENTER OF {APOLLONIUS, SPIEKER, BEVAN} CIRCLES (4.4541, 0.0162) X_2500 = RADICAL CENTER OF {CIRCUMCIRCLE, SPIEKER CIRCLE, APOLLONIUS CIRCLE} (3.4811, 0.5027)
A
B
C
X512
X650
X522
X2490
X513
X2532
X2500
(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) { 512, 650 }: Central Line L(512, 650) { 522, 2490 }: Central Line L(522, 2490) { 513, 2532 }: Central Line L(513, 2532)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0565, -0.0893) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (7.5663, -1.2021) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.0316, 0.2303) X_2490 = RADICAL CENTER OF {CIRCUMCIRCLE,NINE-POINT-CIRCLE, SPIEKER CIRCLE} (-2.7661, 1.3080) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_2532 = RADICAL CENTER OF {APOLLONIUS, SPIEKER, BEVAN} CIRCLES (5.2584, 0.0010) X_2500 = RADICAL CENTER OF {CIRCUMCIRCLE, SPIEKER CIRCLE, APOLLONIUS CIRCLE} (4.0894, 0.4872)
A
B
C
X512
X650
X522
X2490
X513
X2532
X2500
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) { 512, 650 }: Central Line L(512, 650) { 522, 2490 }: Central Line L(522, 2490) { 513, 2532 }: Central Line L(513, 2532)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_512 = ISOGONAL CONJUGATE OF X(99) (1.1621, -0.1275) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (3.4560, -0.1300) X_522 = ISOGONAL CONJUGATE OF X(109) (-0.1115, 0.1100) X_2490 = RADICAL CENTER OF {CIRCUMCIRCLE,NINE-POINT-CIRCLE, SPIEKER CIRCLE} (-0.8985, 0.3951) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_2532 = RADICAL CENTER OF {APOLLONIUS, SPIEKER, BEVAN} CIRCLES (2.6101, 0.2015) X_2500 = RADICAL CENTER OF {CIRCUMCIRCLE, SPIEKER CIRCLE, APOLLONIUS CIRCLE} (2.2077, 0.3162)
A
B
C
X512
X650
X522
X2490
X513
X2532
X2500
Kimberling (6, 9, 13) | AB = 1
X2500 (RADICAL CENTER OF {CIRCUMCIRCLE, SPIEKER CIRCLE, APOLLONIUS 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 \(f(a,b,c) = (b - c)[4bc(b + c)^{2} - 2a^{2}(b^{2} + c^{2}) + a(b + c)(3b^{2} - 2bc + 3c^{2}) - 5a^{3}(b + c)]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-12.793886821580\)
▸ Incident Central Lines3 lines
