X2511
RADICAL CENTER OF {NINE-POINT, APOLLONIUS, MOSES} CIRCLES
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) { 44, 513 }: Central Line L(44, 513) { 115, 804 }: Central Line L(115, 804) { 665, 1213 }: Central Line L(665, 1213) { 900, 2092 }: Central Line L(900, 2092)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_804 = ODD (- 2, 2) INFINITY POINT (0.9171, 0.6474) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.4579, 1.0988) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5294, 0.1914) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (0.9566, 0.6107) X_2092 = DANNEELS-APOLLONIUS PERSPECTOR (0.5833, 0.2396) X_2511 = RADICAL CENTER OF {NINE-POINT, APOLLONIUS, MOSES} CIRCLES (2.3606, 1.9810)
A
B
C
X44
X513
X115
X804
X665
X1213
X900
X2092
X2511
(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) { 44, 513 }: Central Line L(44, 513) { 115, 804 }: Central Line L(115, 804) { 665, 1213 }: Central Line L(665, 1213) { 900, 2092 }: Central Line L(900, 2092)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7445, 0.0609) X_804 = ODD (- 2, 2) INFINITY POINT (0.9585, 0.5612) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.1915, 0.6383) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5345, 0.1495) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (1.0067, 0.5050) X_2092 = DANNEELS-APOLLONIUS PERSPECTOR (0.6202, 0.2061) X_2511 = RADICAL CENTER OF {NINE-POINT, APOLLONIUS, MOSES} CIRCLES (1.8073, 1.0964)
A
B
C
X44
X513
X115
X804
X665
X1213
X900
X2092
X2511
(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) { 44, 513 }: Central Line L(44, 513) { 115, 804 }: Central Line L(115, 804) { 665, 1213 }: Central Line L(665, 1213) { 900, 2092 }: Central Line L(900, 2092)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7135, 0.0464) X_804 = ODD (- 2, 2) INFINITY POINT (0.9914, 0.5173) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.3090, 0.6360) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5291, 0.1514) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (1.0295, 0.4625) X_2092 = DANNEELS-APOLLONIUS PERSPECTOR (0.6010, 0.2080) X_2511 = RADICAL CENTER OF {NINE-POINT, APOLLONIUS, MOSES} CIRCLES (2.0461, 1.0940)
A
B
C
X44
X513
X115
X804
X665
X1213
X900
X2092
X2511
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) { 44, 513 }: Central Line L(44, 513) { 115, 804 }: Central Line L(115, 804) { 665, 1213 }: Central Line L(665, 1213) { 900, 2092 }: Central Line L(900, 2092)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_115 = CENTER OF KIEPERT HYPERBOLA (0.6909, 0.0207) X_804 = ODD (- 2, 2) INFINITY POINT (1.1348, 0.3793) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.0594, 0.2647) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5287, 0.0834) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (1.1671, 0.2997) X_2092 = DANNEELS-APOLLONIUS PERSPECTOR (0.6455, 0.1325) X_2511 = RADICAL CENTER OF {NINE-POINT, APOLLONIUS, MOSES} CIRCLES (1.5196, 0.4221)
A
B
C
X44
X513
X115
X804
X665
X1213
X900
X2092
X2511
Kimberling (6, 9, 13) | AB = 1
X2511 (RADICAL CENTER OF {NINE-POINT, APOLLONIUS, MOSES} CIRCLES ) 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 \(f(a,b,c) = (b - c)(b + c)^{2}[a^{4} - a^{2}(b - c)^{2} - b^{2}c^{2}]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-8.460325009112\)
▸ Incident Central Lines5 lines
{44 513 115 804 665 1213 900 2092
