X1409
X71 -BETH CONJUGATE OF X71
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, 19 }: Central Line L(6, 19) { 63, 77 }: Central Line L(63, 77) { 71, 73 }: Central Line L(71, 73) { 31, 184 }: Central Line L(31, 184) { 109, 284 }: Central Line L(109, 284) { 48, 577 }: Central Line L(48, 577)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4807, 0.0915) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_19 = CLAWSON POINT (0.6246, 0.4244) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_77 = ISOGONAL CONJUGATE OF X(33) (0.5270, 0.1305) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_284 = -CROSS CONJUGATE OF X(21) (0.5646, 0.2088) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_577 = HARMONIC OF X(216) (0.4663, 0.0464) X_1409 = -BETH CONJUGATE OF X(71) (0.5848, 0.1701)
A
B
C
X71
X6
X19
X63
X77
X73
X31
X184
X109
X284
X48
X577
X1409
(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, 19 }: Van Aubel Line { 63, 77 }: Central Line L(63, 77) { 71, 73 }: Central Line L(71, 73) { 31, 184 }: Central Line L(31, 184) { 109, 284 }: Central Line L(109, 284) { 48, 577 }: Central Line L(48, 577)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4706, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_19 = CLAWSON POINT (0.6400, 0.4800) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4286, 0.0000) X_77 = ISOGONAL CONJUGATE OF X(33) (0.5294, 0.0000) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5714, 0.0000) X_31 = 2nd POWER POINT (0.6667, 0.2778) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.6400, 0.0000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_284 = -CROSS CONJUGATE OF X(21) (0.5946, 0.1622) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5714, 0.0000) X_577 = HARMONIC OF X(216) (0.5000, 0.0000) X_1409 = -BETH CONJUGATE OF X(71) (0.6400, 0.0000)
A
B
C
X71
X6
X19
X63
X77
X73
X31
X184
X109
X284
X48
X577
X1409
(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, 19 }: Van Aubel Line { 63, 77 }: Central Line L(63, 77) { 71, 73 }: Central Line L(71, 73) { 31, 184 }: Central Line L(31, 184) { 109, 284 }: Central Line L(109, 284) { 48, 577 }: Central Line L(48, 577)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4753, 0.0000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_19 = CLAWSON POINT (0.6180, 0.4859) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4401, 0.0000) X_77 = ISOGONAL CONJUGATE OF X(33) (0.5247, 0.0000) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5599, 0.0000) X_31 = 2nd POWER POINT (0.6411, 0.2822) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.6180, 0.0000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_284 = -CROSS CONJUGATE OF X(21) (0.5794, 0.1635) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5599, 0.0000) X_577 = HARMONIC OF X(216) (0.5000, 0.0000) X_1409 = -BETH CONJUGATE OF X(71) (0.6180, 0.0000)
A
B
C
X71
X6
X19
X63
X77
X73
X31
X184
X109
X284
X48
X577
X1409
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, 19 }: Central Line L(6, 19) { 63, 77 }: Central Line L(63, 77) { 71, 73 }: Central Line L(71, 73) { 31, 184 }: Central Line L(31, 184) { 109, 284 }: Central Line L(109, 284) { 48, 577 }: Central Line L(48, 577)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4699, -0.1616) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_19 = CLAWSON POINT (0.5737, 0.4687) X_63 = ISOGONAL CONJUGATE OF X(19) (0.3991, -0.1098) X_77 = ISOGONAL CONJUGATE OF X(33) (0.7007, 0.6477) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.6250, 0.5094) X_31 = 2nd POWER POINT (0.6747, 0.1958) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.3215, 1.0549) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_284 = -CROSS CONJUGATE OF X(21) (0.6229, 0.0691) X_48 = CROSSPOINT OF X(1) AND X(63) (0.8073, -1.4298) X_577 = HARMONIC OF X(216) (0.6307, 0.0911) X_1409 = -BETH CONJUGATE OF X(71) (0.5965, 0.3861)
A
B
C
X71
X6
X19
X63
X77
X73
X31
X184
X109
X284
X48
X577
X1409
Kimberling (6, 9, 13) | AB = 1
X1409 (X71 -BETH CONJUGATE OF X71 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a(\sec B + \sec C) : :\)
Search Value (6, 9, 13): \(-0.806655598856\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines11 lines
{6 19 31 184 48 577 63 77 71 73 109 284 213 1042 217 1404 287 651 800 1195 1333 1399
▸ Functional Relations13 relations
X1409 isogonal conjugate of X31623
X1409 complement of anticomplementary conjugate of X18667
X1409 anticomplement of complementary conjugate of X18592
X1409 X63 X1896
X1409 crossdifference of every pair of points on line X521 X1948
X1409 i -Ceva conjugate of Xj for these (i,j): (6 1400 65 1402 73 228 222 73
X1409 crosspoint of Xi and Xj for these (i,j): (6 48 65 1214 222 603
X1409 crosssum of Xi and Xj for these (i,j): (2 92 21 1172 281 318
X1409 X92 X21
X1409
X1409 X1946
X1409
X1409 isotomic conjugate of X22341
Notes
Let V = isotomic conjugate of polar conjugate of line X1 X3 X1 X3 X1409 Randy Hutson , December 26, 2015)
