X693
ISOTOMIC CONJUGATE OF X100
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) { 2, 650 }: Central Line L(2, 650) { 320, 350 }: Central Line L(320, 350) { 325, 523 }: Central Line L(325, 523) { 514, 661 }: Central Line L(514, 661) { 76, 764 }: Central Line L(76, 764) { 100, 927 }: Central Line L(100, 927)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_2 = CENTROID (0.5374, 0.2000) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_320 = ISOTOMIC CONJUGATE OF X(80) (-2.6735, -3.5993) X_350 = -HIRST INVERSE OF X(75) (-3.5227, -3.1370) X_325 = -HIRST INVERSE OF X(69) (-0.3370, -0.7398) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_764 = TRILINEAR CUBE OF X(513) (1.2143, -0.1166) X_927 = ISOGONAL CONJUGATE OF X(926) (1.0037, 0.1836) X_693 = ISOTOMIC CONJUGATE OF X(100) (-25.8163, 8.9981)
A
B
C
X100
X2
X650
X320
X350
X325
X523
X514
X661
X76
X764
X927
X693
(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) { 2, 650 }: Central Line L(2, 650) { 320, 350 }: Central Line L(320, 350) { 325, 523 }: Central Line L(325, 523) { 514, 661 }: Central Line L(514, 661) { 76, 764 }: Central Line L(76, 764) { 100, 927 }: Central Line L(100, 927)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_2 = CENTROID (0.5467, 0.1600) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (6.4000, -1.2000) X_320 = ISOTOMIC CONJUGATE OF X(80) (-1.5600, -1.9200) X_350 = -HIRST INVERSE OF X(75) (-2.3437, -1.5282) X_325 = -HIRST INVERSE OF X(69) (-0.0578, -0.3581) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0987, -0.0010) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0889, -0.0314) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (20.8000, -8.4000) X_76 = 3rd BROCARD POINT (0.4124, 0.0899) X_764 = TRILINEAR CUBE OF X(513) (1.2000, -0.1000) X_927 = ISOGONAL CONJUGATE OF X(926) (0.9394, 0.2386) X_693 = ISOTOMIC CONJUGATE OF X(100) (-11.1600, 2.8800)
A
B
C
X100
X2
X650
X320
X350
X325
X523
X514
X661
X76
X764
X927
X693
(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) { 2, 650 }: Central Line L(2, 650) { 320, 350 }: Central Line L(320, 350) { 325, 523 }: Central Line L(325, 523) { 514, 661 }: Central Line L(514, 661) { 76, 764 }: Central Line L(76, 764) { 100, 927 }: Central Line L(100, 927)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_2 = CENTROID (0.5393, 0.1620) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (7.5663, -1.2021) X_320 = ISOTOMIC CONJUGATE OF X(80) (-1.4042, -2.1471) X_350 = -HIRST INVERSE OF X(75) (-2.2720, -1.7862) X_325 = -HIRST INVERSE OF X(69) (0.0000, -0.3931) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0981, 0.0262) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0910, -0.0001) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (24.7569, -8.3524) X_76 = 3rd BROCARD POINT (0.4271, 0.0928) X_764 = TRILINEAR CUBE OF X(513) (1.2380, -0.0699) X_927 = ISOGONAL CONJUGATE OF X(926) (0.8931, 0.3090) X_693 = ISOTOMIC CONJUGATE OF X(100) (-13.5145, 2.8901)
A
B
C
X100
X2
X650
X320
X350
X325
X523
X514
X661
X76
X764
X927
X693
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) { 2, 650 }: Central Line L(2, 650) { 320, 350 }: Central Line L(320, 350) { 325, 523 }: Central Line L(325, 523) { 514, 661 }: Central Line L(514, 661) { 76, 764 }: Central Line L(76, 764) { 100, 927 }: Central Line L(100, 927)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_2 = CENTROID (0.5444, 0.0934) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (3.4560, -0.1300) X_320 = ISOTOMIC CONJUGATE OF X(80) (-0.5706, -0.8023) X_350 = -HIRST INVERSE OF X(75) (-1.3735, -0.5734) X_325 = -HIRST INVERSE OF X(69) (0.2079, -0.1215) X_523 = ISOGONAL CONJUGATE OF X(110) (1.1957, 0.0148) X_514 = ISOGONAL CONJUGATE OF X(101) (1.1917, -0.0133) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (9.8407, -1.9504) X_76 = 3rd BROCARD POINT (0.3495, 0.0360) X_764 = TRILINEAR CUBE OF X(513) (1.2180, -0.0418) X_927 = ISOGONAL CONJUGATE OF X(926) (0.7313, 0.2468) X_693 = ISOTOMIC CONJUGATE OF X(100) (-5.2790, 0.5401)
A
B
C
X100
X2
X650
X320
X350
X325
X523
X514
X661
X76
X764
X927
X693
Kimberling (6, 9, 13) | AB = 1
X693 (ISOTOMIC CONJUGATE OF X100 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((b - c)/a^{2} : (c - a)/b^{2} : (a - b)/c^{2}\)
Barycentrics: \((b - c)/a : (c - a)/b : (a - b)/c\)
Search Value (6, 9, 13): \(43.948021245883\)
▸ Alternate Coordinates2 forms
▸ Incident Central Lines8 lines
{2 650 76 764 100 927 320 350 321 824 325 523 514 661 649 812
▸ Functional Relations14 relations
X693 isogonal conjugate of X692
X693 X100
X693 complement of X17494
X693 anticomplement of X650
X693 cevapoint of X2 X149
X693 i -cross conjugate of Xj for these (i,j): (11 2 523 514
X693 crosssum of Xi and Xj for these (i,j): (31 667 42 663 649 1475
X693 crossdifference of every pair of points on line X31 X32
X693 trilinear pole of line X918 X1086
X693 trilinear polar of X1783 X19 X25
X693 X1783
X693 X6 isoconjugate of X101
X693 22160
X693
Notes
X693 A29 B29 C29 and A30 B30 C30 be Gemini triangles 29 and 30, resp. Let E29 and E30 be the {ABC , Gemini 29}-circumconic and {ABC , Gemini 30}-circumconic, resp. Let A' be the intersection of the tangent to E29 at A29 and the tangent to E30 at A30 . Define B' and C' cyclically. The lines AA' , BB' , CC″ concur in X693 Randy Hutson , January 15, 2019)
