X1491
CROSSDIFFERENCE OF X1 AND X32
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) { 659, 650 }: Central Line L(659, 650) { 10, 514 }: Central Line L(10, 514) { 44, 513 }: Central Line L(44, 513) { 325, 523 }: Central Line L(325, 523) { 667, 830 }: Central Line L(667, 830)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_659 = CROSSDIFFERENCE OF X(1) AND X(39) (1.9286, 2.2162) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_325 = -HIRST INVERSE OF X(69) (-0.3370, -0.7398) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_667 = CROSSDIFFERENCE OF X(2) AND X(37) (-6.1667, 5.7155) X_830 = ODD (1, 3) INFINITY POINT (1.0590, -0.0690) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (25.5000, -10.6145)
A
B
C
X659
X650
X10
X514
X44
X513
X325
X523
X667
X830
X1491
(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) { 659, 650 }: Central Line L(659, 650) { 10, 514 }: Central Line L(10, 514) { 44, 513 }: Central Line L(44, 513) { 325, 523 }: Central Line L(325, 523) { 667, 830 }: Central Line L(667, 830)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_659 = CROSSDIFFERENCE OF X(1) AND X(39) (1.4000, 1.3000) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (6.4000, -1.2000) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0889, -0.0314) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_325 = -HIRST INVERSE OF X(69) (-0.0578, -0.3581) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0987, -0.0010) X_667 = CROSSDIFFERENCE OF X(2) AND X(37) (-2.0000, 2.5000) X_830 = ODD (1, 3) INFINITY POINT (1.0685, -0.0815) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (11.4000, -3.7000)
A
B
C
X659
X650
X10
X514
X44
X513
X325
X523
X667
X830
X1491
(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) { 659, 650 }: Central Line L(659, 650) { 10, 514 }: Central Line L(10, 514) { 44, 513 }: Central Line L(44, 513) { 325, 523 }: Central Line L(325, 523) { 667, 830 }: Central Line L(667, 830)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_659 = CROSSDIFFERENCE OF X(1) AND X(39) (1.6180, 1.2720) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (7.5663, -1.2021) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0910, -0.0001) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_325 = -HIRST INVERSE OF X(69) (0.0000, -0.3931) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0981, 0.0262) X_667 = CROSSDIFFERENCE OF X(2) AND X(37) (-2.4741, 2.4741) X_830 = ODD (1, 3) INFINITY POINT (1.0760, -0.0445) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (13.5145, -3.6762)
A
B
C
X659
X650
X10
X514
X44
X513
X325
X523
X667
X830
X1491
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) { 659, 650 }: Central Line L(659, 650) { 10, 514 }: Central Line L(10, 514) { 44, 513 }: Central Line L(44, 513) { 325, 523 }: Central Line L(325, 523) { 667, 830 }: Central Line L(667, 830)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_659 = CROSSDIFFERENCE OF X(1) AND X(39) (1.1270, 0.5340) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (3.4560, -0.1300) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_514 = ISOGONAL CONJUGATE OF X(101) (1.1917, -0.0133) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_325 = -HIRST INVERSE OF X(69) (0.2079, -0.1215) X_523 = ISOGONAL CONJUGATE OF X(110) (1.1957, 0.0148) X_667 = CROSSDIFFERENCE OF X(2) AND X(37) (-0.3954, 0.7848) X_830 = ODD (1, 3) INFINITY POINT (1.1800, -0.0690) X_1491 = CROSSDIFFERENCE OF X(1) AND X(32) (5.7851, -0.7941)
A
B
C
X659
X650
X10
X514
X44
X513
X325
X523
X667
X830
X1491
Kimberling (6, 9, 13) | AB = 1
X1491 (CROSSDIFFERENCE OF X1 AND X32 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(b^{3} - c^{3} : :\)
Search Value (6, 9, 13): \(-29.540153474797\)
▸ Incident Central Lines5 lines
▸ Functional Relations8 relations
X1491 X659 X650
X1491 isogonal conjugate of X1492
X1491 isotomic conjugate of X789
X1491 i -Ceva conjugate of Xj for these (i,j): (262 11 1492 1
X1491 cevapoint of X1 X1492
X1491 crosspoint of Xi and Xj for these (i,j): (1 1492
X1491 crosssum of Xi and Xj for these (i,j): (1 1491 100 1390
X1491 crossdifference of every pair of points on line X1 X32
Notes
Let L be the line PU(10) = X10 X514 trilinear polar of the cevapoint of PU(10), so that M = X522 X1491 L , M, and VW concur in X1491 Randy Hutson , December 26, 2015)
