X901
ISOGONAL CONJUGATE OF X900
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) { 953, 3 }: Central Line L(953, 3) { 36, 106 }: Central Line L(36, 106) { 59, 109 }: Central Line L(59, 109) { 88, 105 }: Central Line L(88, 105) { 100, 513 }: Central Line L(100, 513)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0000, 0.2041) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_59 = ISOGONAL CONJUGATE OF X(11) (0.5333, 0.4355) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_901 = ISOGONAL CONJUGATE OF X(900) (1.0000, -0.0000)
A
B
C
X953
X3
X36
X106
X59
X109
X88
X105
X100
X513
X901
(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) { 953, 3 }: Central Line L(953, 3) { 36, 106 }: Central Line L(36, 106) { 59, 109 }: Central Line L(59, 109) { 88, 105 }: Central Line L(88, 105) { 100, 513 }: Central Line L(100, 513)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0000, 0.0000) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.0000, 1.0000) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_59 = ISOGONAL CONJUGATE OF X(11) (0.6000, 0.4000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9800, -0.1400) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_901 = ISOGONAL CONJUGATE OF X(900) (1.0000, -0.0000)
A
B
C
X953
X3
X36
X106
X59
X109
X88
X105
X100
X513
X901
(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) { 953, 3 }: Central Line L(953, 3) { 36, 106 }: Central Line L(36, 106) { 59, 109 }: Central Line L(59, 109) { 88, 105 }: Central Line L(88, 105) { 100, 513 }: Central Line L(100, 513)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0134, -0.1152) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.9387, 1.0546) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) X_59 = ISOGONAL CONJUGATE OF X(11) (0.5932, 0.4263) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9710, -0.0490) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9563, -0.2045) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_901 = ISOGONAL CONJUGATE OF X(900) (0.9866, 0.1152)
A
B
C
X953
X3
X36
X106
X59
X109
X88
X105
X100
X513
X901
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) { 953, 3 }: Central Line L(953, 3) { 36, 106 }: Central Line L(36, 106) { 59, 109 }: Central Line L(59, 109) { 88, 105 }: Central Line L(88, 105) { 100, 513 }: Central Line L(100, 513)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0620, -0.6402) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.7120, 0.4690) X_106 = Λ(INCENTER, CENTROID) (1.0294, -0.0621) X_59 = ISOGONAL CONJUGATE OF X(11) (0.6302, 0.2712) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9802, -0.0393) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (1.0692, -0.3135) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_901 = ISOGONAL CONJUGATE OF X(900) (0.9380, 0.0908)
A
B
C
X953
X3
X36
X106
X59
X109
X88
X105
X100
X513
X901
Kimberling (6, 9, 13) | AB = 1
X901 (ISOGONAL CONJUGATE OF X900 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a/[(b - c)(b + c - 2a)] : :\)
Search Value (6, 9, 13): \(-0.449531897666\)
X901
▸ Incident Central Lines11 lines
{3 953 6 2384 36 106 55 840 59 109 88 105 100 513 101 649 104 517 484 759 675 903
▸ Functional Relations21 relations
Notes
Let LA be the line of reflection of line X1 X5 BC , and define LB and LC cyclically. Let A' = LB ∩LC , and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X901 Randy Hutson , 9/23/2011)
Let La , Lb , Lc be the lines through A , B , C , resp. parallel to the antiorthic axis. Let Ma , Mb , Mc be the reflections of BC , CA , AB in La , Lb , Lc , resp. Let A' = Mb ∩Mc , and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC . The centroid of A'B'C' is X901 Hyacinthos #16741 /16782, Sep 2008. (Randy Hutson, March 25, 2016)
Let A' , B' , C' be the intersections of the Nagel line and lines BC , CA , AB , resp. The circumcircles of AB'C' , BC'A' , CA'B' concur in X901
