X2393
ISOGONAL CONJUGATE OF X2373
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) { 6, 25 }: Central Line L(6, 25) { 66, 69 }: Central Line L(66, 69) { 160, 216 }: Central Line L(160, 216) { 30, 511 }: Central Line L(30, 511) { 157, 577 }: Central Line L(157, 577) { 23, 895 }: Central Line L(23, 895)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2373 = CIR(X(67)) (0.5684, -0.4036) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5912, 1.0593) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_160 = -OF-TANGENTIAL-TRIANGLE (0.4841, -0.0647) X_216 = -CEVA CONJUGATE OF X(51) (0.5201, 0.1353) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5266, 0.3812) X_577 = HARMONIC OF X(216) (0.4663, 0.0464) X_23 = FAR-OUT POINT (1.3871, 2.4232) X_895 = ISOGONAL CONJUGATE OF X(468) (0.9484, -0.0134) X_2393 = ISOGONAL CONJUGATE OF X(2373) (0.6414, 0.7775)
A
B
C
X2373
X6
X25
X66
X69
X160
X216
X30
X511
X157
X577
X23
X895
X2393
(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) { 6, 25 }: Van Aubel Line { 66, 69 }: Central Line L(66, 69) { 160, 216 }: Central Line L(160, 216) { 30, 511 }: Central Line L(30, 511) { 157, 577 }: Central Line L(157, 577) { 23, 895 }: Central Line L(23, 895)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2373 = CIR(X(67)) (0.3600, -0.4800) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6400, 0.4800) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5467, -0.4150) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_160 = -OF-TANGENTIAL-TRIANGLE (0.5000, -0.5208) X_216 = -CEVA CONJUGATE OF X(51) (0.5000, 0.0000) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2569, -0.3367) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5000, 0.5208) X_577 = HARMONIC OF X(216) (0.5000, 0.0000) X_23 = FAR-OUT POINT (0.9200, 1.4400) X_895 = ISOGONAL CONJUGATE OF X(468) (0.9200, 0.0000) X_2393 = ISOGONAL CONJUGATE OF X(2373) (0.5467, 0.7350)
A
B
C
X2373
X6
X25
X66
X69
X160
X216
X30
X511
X157
X577
X23
X895
X2393
(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) { 6, 25 }: Van Aubel Line { 66, 69 }: Central Line L(66, 69) { 160, 216 }: Central Line L(160, 216) { 30, 511 }: Central Line L(30, 511) { 157, 577 }: Central Line L(157, 577) { 23, 895 }: Central Line L(23, 895)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2373 = CIR(X(67)) (0.3820, -0.4859) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6180, 0.4859) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5393, -0.4130) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_160 = -OF-TANGENTIAL-TRIANGLE (0.5000, -0.5145) X_216 = -CEVA CONJUGATE OF X(51) (0.5000, 0.0000) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2881, -0.3552) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5000, 0.5145) X_577 = HARMONIC OF X(216) (0.5000, 0.0000) X_23 = FAR-OUT POINT (0.8541, 1.4576) X_895 = ISOGONAL CONJUGATE OF X(468) (0.8541, 0.0000) X_2393 = ISOGONAL CONJUGATE OF X(2373) (0.5393, 0.7370)
A
B
C
X2373
X6
X25
X66
X69
X160
X216
X30
X511
X157
X577
X23
X895
X2393
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) { 6, 25 }: Central Line L(6, 25) { 66, 69 }: Central Line L(66, 69) { 160, 216 }: Central Line L(160, 216) { 30, 511 }: Central Line L(30, 511) { 157, 577 }: Central Line L(157, 577) { 23, 895 }: Central Line L(23, 895)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2373 = CIR(X(67)) (0.0334, -0.6030) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.5783, 0.3748) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5104, -0.5592) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_160 = -OF-TANGENTIAL-TRIANGLE (0.4566, 0.9160) X_216 = -CEVA CONJUGATE OF X(51) (0.6975, 0.2779) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_511 = ISOGONAL CONJUGATE OF X(98) (0.3235, -0.5244) X_157 = -OF-TANGENTIAL-TRIANGLE (0.4713, 0.5134) X_577 = HARMONIC OF X(216) (0.6307, 0.0911) X_23 = FAR-OUT POINT (0.6051, 0.5970) X_895 = ISOGONAL CONJUGATE OF X(468) (0.8172, 0.0352) X_2393 = ISOGONAL CONJUGATE OF X(2373) (0.3126, 0.7071)
A
B
C
X2373
X6
X25
X66
X69
X160
X216
X30
X511
X157
X577
X23
X895
X2393
Kimberling (6, 9, 13) | AB = 1
X2393 (ISOGONAL CONJUGATE OF X2373 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/U : 1/V : 1/W\) , where
\(U : V : W\) =
X2373 Barycentrics: \(a^{2} (a^{4} b^{2} + a^{4} c^{2} - 2 a^{2} b^{2} c^{2} - b^{6} + b^{4} c^{2} + b^{2} c^{4} - c^{6}) : :\)
▸ Incident Central Lines10 lines
{6 25 23 895 30 511 66 69 67 1205 141 1368 157 577 160 216 401 1632 599 1853
▸ Functional Relations5 relations
X2393 X2373
X2393 i -Ceva conjugate of Xj for these (i,j): (4 1560 23 187 895 6 935 647
X2393 crosspoint of Xi and Xj for these (i,j): (4 111 6 67
X2393 crosssum of Xi and Xj for these (i,j): (2 23 3 524 669 1648
X2393 X3 X6 X25
Notes
Let A″B″C″ be the 2nd Ehrmann triangle. Let Pa be the pole of line B″C″ wrt the A-Ehrmann circle, and define Pb and Pc cyclically. The lines A”Pa , B ”Pb , C ”Pc concur in X2393 Randy Hutson , November 18, 2015)
