X1146
CENTER OF HYPERBOLA {{A,B,C,X2 ,X8 }}
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) { 2, 1121 }: Central Line L(2, 1121) { 9, 80 }: Central Line L(9, 80) { 8, 220 }: Central Line L(8, 220) { 115, 124 }: Central Line L(115, 124) { 6, 281 }: Central Line L(6, 281)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_1121 = -BLAIKIE TRANSFORM OF X(2) (0.8358, -0.1143) X_9 = MITTENPUNKT (0.5110, 0.1615) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_8 = NAGEL POINT (0.4694, 0.1333) X_220 = -CEVA CONJUGATE OF X(55) (0.4826, 0.1278) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_124 = -OF-MEDIAL-TRIANGLE (0.6243, 0.0030) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_281 = -CROSS CONJUGATE OF X(9) (0.5796, 0.3202) X_1146 = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}} (0.6866, 0.0428)
A
B
C
X2
X1121
X9
X80
X8
X220
X115
X124
X6
X281
X1146
(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) { 2, 1121 }: Central Line L(2, 1121) { 9, 80 }: Central Line L(9, 80) { 8, 220 }: Central Line L(8, 220) { 115, 124 }: Central Line L(115, 124) { 6, 281 }: Van Aubel Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_1121 = -BLAIKIE TRANSFORM OF X(2) (0.7333, -0.1231) X_9 = MITTENPUNKT (0.5091, 0.1091) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8400, -0.1200) X_8 = NAGEL POINT (0.4400, 0.0800) X_220 = -CEVA CONJUGATE OF X(55) (0.4706, 0.0706) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7445, 0.0609) X_124 = -OF-MEDIAL-TRIANGLE (0.5700, -0.0100) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_281 = -CROSS CONJUGATE OF X(9) (0.6400, 0.4800) X_1146 = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}} (0.6400, 0.0185)
A
B
C
X2
X1121
X9
X80
X8
X220
X115
X124
X6
X281
X1146
(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) { 2, 1121 }: Central Line L(2, 1121) { 9, 80 }: Central Line L(9, 80) { 8, 220 }: Central Line L(8, 220) { 115, 124 }: Central Line L(115, 124) { 6, 281 }: Van Aubel Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_1121 = -BLAIKIE TRANSFORM OF X(2) (0.6967, -0.1352) X_9 = MITTENPUNKT (0.5077, 0.1103) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.7953, -0.1425) X_8 = NAGEL POINT (0.4499, 0.0817) X_220 = -CEVA CONJUGATE OF X(55) (0.4753, 0.0714) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7135, 0.0464) X_124 = -OF-MEDIAL-TRIANGLE (0.5590, -0.0071) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_281 = -CROSS CONJUGATE OF X(9) (0.6180, 0.4859) X_1146 = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}} (0.6180, 0.0134)
A
B
C
X2
X1121
X9
X80
X8
X220
X115
X124
X6
X281
X1146
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) { 2, 1121 }: Central Line L(2, 1121) { 9, 80 }: Central Line L(9, 80) { 8, 220 }: Central Line L(8, 220) { 115, 124 }: Central Line L(115, 124) { 6, 281 }: Central Line L(6, 281)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2 = CENTROID (0.5444, 0.0934) X_1121 = -BLAIKIE TRANSFORM OF X(2) (0.5457, -0.0911) X_9 = MITTENPUNKT (0.5022, 0.0343) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.6972, -0.1168) X_8 = NAGEL POINT (0.4024, 0.0200) X_220 = -CEVA CONJUGATE OF X(55) (0.4740, 0.0105) X_115 = CENTER OF KIEPERT HYPERBOLA (0.6909, 0.0207) X_124 = -OF-MEDIAL-TRIANGLE (0.5142, -0.0030) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_281 = -CROSS CONJUGATE OF X(9) (0.4813, -0.0921) X_1146 = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}} (0.5450, 0.0011)
A
B
C
X2
X1121
X9
X80
X8
X220
X115
X124
X6
X281
X1146
Kimberling (6, 9, 13) | AB = 1
X1146 (CENTER OF HYPERBOLA {{A,B,C,X2 ,X8 }} ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(bc[(b - c)(b + c - a)]^{2} : :\)
Search Value (6, 9, 13): \(3.577245761969\)
X1146 X2052
▸ Incident Central Lines10 lines
{2 664 6 281 8 220 9 80 101 952 115 124 116 514 169 355 515 910 918 1086
▸ Functional Relations18 relations
X1146 X2 X1121
X1146 i in Xj for these (i,j): of (1565 116 2
X1146 35110 in X2
X1146 isogonal conjugate of X1262
X1146 isotomic conjugate of X1275
X1146 complement of X664
X1146 anticomplement of X17044
X1146 crosspoint of Xi and Xj for these (i,j): (2 522 4 514 9 1021
X1146 crosssum of Xi and Xj for these (i,j): (3 101 6 109 56 1415 57 1020 1407 1461
X1146 X6 X41 X1 X15 X16 X1
X1146 crossdifference of every pair of points on line X109 X692
X1146 X1121
X1146 X522
X1146 trilinear poles of lines passing through X522
X1146 X2 Ceva conjugate of X522
X1146 35110
X1146 medial triangle of line X2 X7
X1146 X522
Notes
Let La , Lb , Lc be the lines through A , B , C , resp. parallel to the Soddy line. Let Ma , Mb , Mc be the reflections of BC , CA , AB in La , Lb , Lc , resp. Let A' = Mb ∩Mc , and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC . Let A″B″C″ be the reflection of A'B'C' in the Soddy line. The triangle A″B″C″ is homothetic to ABC , with center of homothety X1146 Hyacinthos #16741 /16782, Sep 2008. (Randy Hutson , March 25, 2016)
