X345
ISOTOMIC CONJUGATE OF X278
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) { 8, 21 }: Central Line L(8, 21) { 2, 37 }: Central Line L(2, 37) { 22, 100 }: Central Line L(22, 100) { 63, 69 }: Central Line L(63, 69) { 190, 329 }: Central Line L(190, 329) { 219, 332 }: Central Line L(219, 332)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_278 = ISOGONAL CONJUGATE OF X(219) (0.6278, 0.4577) X_8 = NAGEL POINT (0.4694, 0.1333) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_2 = CENTROID (0.5374, 0.2000) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_22 = EXETER POINT (0.4143, -0.1221) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3847, -0.0222) X_219 = -CEVA CONJUGATE OF X(55) (0.4277, 0.0625) X_332 = ISOTOMIC CONJUGATE OF X(225) (0.3595, 0.0434) X_345 = ISOTOMIC CONJUGATE OF X(278) (0.3405, 0.0381)
A
B
C
X278
X8
X21
X2
X37
X22
X100
X63
X69
X190
X329
X219
X332
X345
(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) { 8, 21 }: Central Line L(8, 21) { 2, 37 }: Central Line L(2, 37) { 22, 100 }: Central Line L(22, 100) { 63, 69 }: Central Line L(63, 69) { 190, 329 }: Central Line L(190, 329) { 219, 332 }: Central Line L(219, 332)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_278 = ISOGONAL CONJUGATE OF X(219) (0.6400, 0.4800) X_8 = NAGEL POINT (0.4400, 0.0800) X_21 = SCHIFFLER POINT (0.5368, 0.1263) X_2 = CENTROID (0.5467, 0.1600) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5787, 0.1787) X_22 = EXETER POINT (0.3600, -0.4800) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4286, 0.0000) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_190 = YFF PARABOLIC POINT (0.0933, 0.3200) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3400, -0.1200) X_219 = -CEVA CONJUGATE OF X(55) (0.4000, 0.0000) X_332 = ISOTOMIC CONJUGATE OF X(225) (0.2967, 0.0000) X_345 = ISOTOMIC CONJUGATE OF X(278) (0.2727, 0.0000)
A
B
C
X278
X8
X21
X2
X37
X22
X100
X63
X69
X190
X329
X219
X332
X345
(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) { 8, 21 }: Central Line L(8, 21) { 2, 37 }: Central Line L(2, 37) { 22, 100 }: Central Line L(22, 100) { 63, 69 }: Central Line L(63, 69) { 190, 329 }: Central Line L(190, 329) { 219, 332 }: Central Line L(219, 332)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_278 = ISOGONAL CONJUGATE OF X(219) (0.6180, 0.4859) X_8 = NAGEL POINT (0.4499, 0.0817) X_21 = SCHIFFLER POINT (0.5310, 0.1276) X_2 = CENTROID (0.5393, 0.1620) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5661, 0.1805) X_22 = EXETER POINT (0.3820, -0.4859) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4401, 0.0000) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_190 = YFF PARABOLIC POINT (0.1322, 0.3610) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3648, -0.1231) X_219 = -CEVA CONJUGATE OF X(55) (0.4159, 0.0000) X_332 = ISOTOMIC CONJUGATE OF X(225) (0.3270, 0.0000) X_345 = ISOTOMIC CONJUGATE OF X(278) (0.3056, 0.0000)
A
B
C
X278
X8
X21
X2
X37
X22
X100
X63
X69
X190
X329
X219
X332
X345
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) { 8, 21 }: Central Line L(8, 21) { 2, 37 }: Central Line L(2, 37) { 22, 100 }: Central Line L(22, 100) { 63, 69 }: Central Line L(63, 69) { 190, 329 }: Central Line L(190, 329) { 219, 332 }: Central Line L(219, 332)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_278 = ISOGONAL CONJUGATE OF X(219) (0.6201, 0.3162) X_8 = NAGEL POINT (0.4024, 0.0200) X_21 = SCHIFFLER POINT (0.5385, 0.0448) X_2 = CENTROID (0.5444, 0.0934) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5913, 0.1097) X_22 = EXETER POINT (0.8328, 2.4854) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_63 = ISOGONAL CONJUGATE OF X(19) (0.3991, -0.1098) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_190 = YFF PARABOLIC POINT (0.1548, 0.2119) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3430, -0.1882) X_219 = -CEVA CONJUGATE OF X(55) (0.4333, -0.0239) X_332 = ISOTOMIC CONJUGATE OF X(225) (0.2845, -0.0091) X_345 = ISOTOMIC CONJUGATE OF X(278) (0.2575, -0.0064)
A
B
C
X278
X8
X21
X2
X37
X22
X100
X63
X69
X190
X329
X219
X332
X345
Kimberling (6, 9, 13) | AB = 1
X345 (ISOTOMIC CONJUGATE OF X278 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((\csc A)/(1 - \sec A) : (\csc B)/(1 - \sec B) : (\csc C)/(1 - \sec C)\)
Barycentrics: \(1/(1 - \sec A) : 1/(1 - \sec B) : 1/(1 - \sec C)\)
Search Value (6, 9, 13): \(5.922999174659\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines11 lines
{2 37 8 21 22 100 57 728 63 69 78 1040 190 329 219 332 280 341 304 348 498 1089
▸ Functional Relations7 relations
X345 isogonal conjugate of X608
X345 X278
X345 i -Ceva conjugate of Xj for these (i,j): (304 69 332 78
X345 i -cross conjugate of Xj for these (i,j): (78 69 219 8 306 312
X345 complement of X30699
X345 anticomplement of X3772
X345 X1118
Notes
Let A35 B35 C35 be Gemini triangle 35. Let A' be the perspector of conic {A , B , C ,B35 ,C35 }}, and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X345 Randy Hutson , January 15, 2019)
