X3756
X106 com[INVERSE(n(HEXYL TRIANGLE))]
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) { 88, 149 }: Central Line L(88, 149) { 11, 244 }: Central Line L(11, 244) { 106, 952 }: Central Line L(106, 952) { 8, 1120 }: Central Line L(8, 1120) { 1, 1145 }: Central Line L(1, 1145) { 2, 1280 }: Central Line L(2, 1280)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_244 = -LINE CONJUGATE OF X(100) (0.7857, 0.1166) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.1267, 0.6191) X_8 = NAGEL POINT (0.4694, 0.1333) X_1120 = -BLAIKIE TRANSFORM OF X(1) (1.2041, 0.2000) X_1 = INCENTER (0.5714, 0.2333) X_1145 = 3rd EHRMANN POINT (0.3061, 0.2999) X_2 = CENTROID (0.5374, 0.2000) X_1280 = ISOGONAL CONJUGATE OF X(1279) (1.1600, 0.1306) X_3756 = com[INVERSE(n(HEXYL TRIANGLE))] (0.8367, 0.1666)
A
B
C
X88
X149
X11
X244
X106
X952
X8
X1120
X1
X1145
X2
X1280
X3756
(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) { 88, 149 }: Central Line L(88, 149) { 11, 244 }: Central Line L(11, 244) { 106, 952 }: Central Line L(106, 952) { 8, 1120 }: Central Line L(8, 1120) { 1, 1145 }: Central Line L(1, 1145) { 2, 1280 }: Central Line L(2, 1280)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_149 = REFLECTION OF X(20) IN X(104) (1.2400, -0.3200) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_244 = -LINE CONJUGATE OF X(100) (0.8000, 0.1000) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.2017, 0.6200) X_8 = NAGEL POINT (0.4400, 0.0800) X_1120 = -BLAIKIE TRANSFORM OF X(1) (1.3200, 0.2400) X_1 = INCENTER (0.6000, 0.2000) X_1145 = 3rd EHRMANN POINT (0.3200, 0.2400) X_2 = CENTROID (0.5467, 0.1600) X_1280 = ISOGONAL CONJUGATE OF X(1279) (1.2800, 0.1600) X_3756 = com[INVERSE(n(HEXYL TRIANGLE))] (0.8800, 0.1600)
A
B
C
X88
X149
X11
X244
X106
X952
X8
X1120
X1
X1145
X2
X1280
X3756
(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) { 88, 149 }: Central Line L(88, 149) { 11, 244 }: Central Line L(11, 244) { 106, 952 }: Central Line L(106, 952) { 8, 1120 }: Central Line L(8, 1120) { 1, 1145 }: Central Line L(1, 1145) { 2, 1280 }: Central Line L(2, 1280)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9710, -0.0490) X_149 = REFLECTION OF X(20) IN X(104) (1.1407, -0.3666) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_244 = -LINE CONJUGATE OF X(100) (0.7719, 0.0802) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.2388, 0.6521) X_8 = NAGEL POINT (0.4499, 0.0817) X_1120 = -BLAIKIE TRANSFORM OF X(1) (1.2898, 0.1988) X_1 = INCENTER (0.5841, 0.2021) X_1145 = 3rd EHRMANN POINT (0.3443, 0.2540) X_2 = CENTROID (0.5393, 0.1620) X_1280 = ISOGONAL CONJUGATE OF X(1279) (1.2237, 0.1170) X_3756 = com[INVERSE(n(HEXYL TRIANGLE))] (0.8698, 0.1403)
A
B
C
X88
X149
X11
X244
X106
X952
X8
X1120
X1
X1145
X2
X1280
X3756
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) { 88, 149 }: Central Line L(88, 149) { 11, 244 }: Central Line L(11, 244) { 106, 952 }: Central Line L(106, 952) { 8, 1120 }: Central Line L(8, 1120) { 1, 1145 }: Central Line L(1, 1145) { 2, 1280 }: Central Line L(2, 1280)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9802, -0.0393) X_149 = REFLECTION OF X(20) IN X(104) (0.9920, -0.2537) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_244 = -LINE CONJUGATE OF X(100) (0.7876, 0.0501) X_106 = Λ(INCENTER, CENTROID) (1.0294, -0.0621) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.3380, 0.7161) X_8 = NAGEL POINT (0.4024, 0.0200) X_1120 = -BLAIKIE TRANSFORM OF X(1) (1.5416, 0.2024) X_1 = INCENTER (0.6154, 0.1300) X_1145 = 3rd EHRMANN POINT (0.3615, 0.1434) X_2 = CENTROID (0.5444, 0.0934) X_1280 = ISOGONAL CONJUGATE OF X(1279) (1.6028, 0.1375) X_3756 = com[INVERSE(n(HEXYL TRIANGLE))] (0.9720, 0.1112)
A
B
C
X88
X149
X11
X244
X106
X952
X8
X1120
X1
X1145
X2
X1280
X3756
Kimberling (6, 9, 13) | AB = 1
X3756 (X106 com[INVERSE(n(HEXYL TRIANGLE))] ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where \(f(a,b,c) = (3a - b - c)(b - c)^{2}\)
Search Value (6, 9, 13): \(-0.928012514996\)
▸ Linear Combinations1 combination
▸ Incident Central Lines10 lines
{1 1145 2 1280 8 1120 11 244 88 149 106 952 513 1357 528 1054 1015 1146 1616 1788
▸ Functional Relations2 relations
Notes
Let La , Lb , Lc be the lines through A , B , C , resp. parallel to the Nagel line. Let Ma , Mb , Mc be the reflections of BC , CA , AB in La , Lb , Lc , resp. Let A' = Mb ∩Mc , B' = Mc ∩Ma , C' = Ma ∩Mb . 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 Nagel line. The triangle A″B″C″ is homothetic to ABC , with center of homothety X3756 Hyacinthos #16741 /16782, Sep 2008. (Randy Hutson , March 25, 2016)
