X919
ISOGONAL CONJUGATE OF X918
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) { 104, 294 }: Central Line L(104, 294) { 100, 650 }: Central Line L(100, 650) { 109, 649 }: Central Line L(109, 649) { 101, 663 }: Central Line L(101, 663) { 99, 666 }: Central Line L(99, 666) { 103, 672 }: Central Line L(103, 672)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.0192, 0.5351) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_294 = ISOGONAL CONJUGATE OF X(241) (0.9276, -0.0575) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-21.0000, 14.6969) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (8.6667, -3.2660) X_99 = STEINER POINT (0.1109, 0.4322) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.0982, 0.1428) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_672 = CROSSDIFFERENCE OF X(1) AND X(514) (1.9245, 2.2184) X_919 = ISOGONAL CONJUGATE OF X(918) (1.0058, 0.1699)
A
B
C
X918
X104
X294
X100
X650
X109
X649
X101
X663
X99
X666
X103
X672
X919
(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) { 104, 294 }: Central Line L(104, 294) { 100, 650 }: Central Line L(100, 650) { 109, 649 }: Central Line L(109, 649) { 101, 663 }: Central Line L(101, 663) { 99, 666 }: Central Line L(99, 666) { 103, 672 }: Central Line L(103, 672)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.0687, 0.4010) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_294 = ISOGONAL CONJUGATE OF X(241) (0.8883, -0.0579) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (6.4000, -1.2000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-8.0000, 6.0000) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.3077, 0.4615) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (4.0000, -1.0000) X_99 = STEINER POINT (0.1511, 0.3581) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.1272, 0.1723) X_103 = ANTIPODE OF X(101) (0.6923, -0.4615) X_672 = CROSSDIFFERENCE OF X(1) AND X(514) (1.3913, 1.3043) X_919 = ISOGONAL CONJUGATE OF X(918) (0.9561, 0.2049)
A
B
C
X918
X104
X294
X100
X650
X109
X649
X101
X663
X99
X666
X103
X672
X919
(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) { 104, 294 }: Central Line L(104, 294) { 100, 650 }: Central Line L(100, 650) { 109, 649 }: Central Line L(109, 649) { 101, 663 }: Central Line L(101, 663) { 99, 666 }: Central Line L(99, 666) { 103, 672 }: Central Line L(103, 672)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.0745, 0.3723) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_294 = ISOGONAL CONJUGATE OF X(241) (0.8391, -0.0765) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (7.5663, -1.2021) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-9.6244, 5.9482) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.3365, 0.4725) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (4.6762, -1.0000) X_99 = STEINER POINT (0.1910, 0.3931) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.1180, 0.2429) X_103 = ANTIPODE OF X(101) (0.6635, -0.4725) X_672 = CROSSDIFFERENCE OF X(1) AND X(514) (1.3003, 1.4042) X_919 = ISOGONAL CONJUGATE OF X(918) (0.9173, 0.2754)
A
B
C
X918
X104
X294
X100
X650
X109
X649
X101
X663
X99
X666
X103
X672
X919
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) { 104, 294 }: Central Line L(104, 294) { 100, 650 }: Central Line L(100, 650) { 109, 649 }: Central Line L(109, 649) { 101, 663 }: Central Line L(101, 663) { 99, 666 }: Central Line L(99, 666) { 103, 672 }: Central Line L(103, 672)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_918 = ISOTOMIC CONJUGATE OF X(666) (1.1912, 0.2031) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_294 = ISOGONAL CONJUGATE OF X(241) (0.7790, -0.0299) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (3.4560, -0.1300) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-2.9286, 1.6903) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.4482, 0.2934) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (2.1378, -0.1207) X_99 = STEINER POINT (0.2513, 0.2387) X_666 = TRILINEAR POLE OF LINE X(2)X(11) (1.0867, 0.1883) X_103 = ANTIPODE OF X(101) (0.5518, -0.8428) X_672 = CROSSDIFFERENCE OF X(1) AND X(514) (0.9049, 0.5974) X_919 = ISOGONAL CONJUGATE OF X(918) (0.8087, 0.2051)
A
B
C
X918
X104
X294
X100
X650
X109
X649
X101
X663
X99
X666
X103
X672
X919
Kimberling (6, 9, 13) | AB = 1
X919 (ISOGONAL CONJUGATE OF X918 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a^{2}/(w/b - v/c) : :\) , where
\(u : v : w\) =
X100 Search Value (6, 9, 13): \(-0.610000059147\)
X919
▸ Incident Central Lines11 lines
{6 840 99 666 100 650 101 663 103 672 104 294 105 910 106 1055 109 649 673 675 885 929
▸ Functional Relations8 relations
X919 X918
X919 trilinear pole of line X6 X692
X919 i , Xj ) for these (i,j): (1 41 2 11 6 692 11 650 76 8
X919 isotomic conjugate of trilinear pole of line X2 X11
X919 A , B , C , of conic {A , B , C ,PU(95)}}
X919 X100
X919 X101 X105 X1
X919 X100 X105 X2
Notes
Let Q be a point on line X2 X11 X2 A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC ∩B'C' ), and define B″ , C″ cyclically. The circumcircles of AB″C″ , BC″A″ , CA″B″ concur in X919 Randy Hutson , February 10, 2016)
Let A' , B' , C' be the intersections of line X1 X6 BC , CA , AB , resp. The circumcircles of AB'C' , BC'A' , CA'B' concur in X919
