X870
ISOGONAL CONJUGATE OF X869
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) { 1, 76 }: Central Line L(1, 76) { 6, 75 }: Central Line L(6, 75) { 56, 85 }: Central Line L(56, 85) { 2, 292 }: Central Line L(2, 292) { 58, 274 }: Central Line L(58, 274) { 34, 331 }: Central Line L(34, 331)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_869 = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32) (0.6039, 0.2651) X_1 = INCENTER (0.5714, 0.2333) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5610, 0.2421) X_2 = CENTROID (0.5374, 0.2000) X_292 = -HIRST INVERSE OF X(291) (1.0127, 0.0417) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5168, 0.1843) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_331 = ISOTOMIC CONJUGATE OF X(219) (0.6062, 0.4046) X_870 = ISOGONAL CONJUGATE OF X(869) (0.5328, 0.2015)
A
B
C
X869
X1
X76
X6
X75
X56
X85
X2
X292
X58
X274
X34
X331
X870
(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) { 1, 76 }: Central Line L(1, 76) { 6, 75 }: Central Line L(6, 75) { 56, 85 }: Central Line L(56, 85) { 2, 292 }: Central Line L(2, 292) { 58, 274 }: Central Line L(58, 274) { 106, 789 }: Central Line L(106, 789)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_869 = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32) (0.6480, 0.2360) X_1 = INCENTER (0.6000, 0.2000) X_76 = 3rd BROCARD POINT (0.4124, 0.0899) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4826, 0.1226) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6667, 0.3333) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5801, 0.2201) X_2 = CENTROID (0.5467, 0.1600) X_292 = -HIRST INVERSE OF X(291) (1.0074, 0.0540) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6522, 0.2609) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5114, 0.1416) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_789 = o-(- 3, 0) CIRCUMCIRCLE POINT (0.1092, 0.3119) X_870 = ISOGONAL CONJUGATE OF X(869) (0.5360, 0.1624)
A
B
C
X869
X1
X76
X6
X75
X56
X85
X2
X292
X58
X274
X106
X789
X870
(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) { 1, 76 }: Central Line L(1, 76) { 6, 75 }: Central Line L(6, 75) { 56, 85 }: Central Line L(56, 85) { 2, 292 }: Central Line L(2, 292) { 58, 274 }: Central Line L(58, 274) { 106, 789 }: Central Line L(106, 789)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_869 = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32) (0.6248, 0.2386) X_1 = INCENTER (0.5841, 0.2021) X_76 = 3rd BROCARD POINT (0.4271, 0.0928) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4859, 0.1249) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6411, 0.3392) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5676, 0.2231) X_2 = CENTROID (0.5393, 0.1620) X_292 = -HIRST INVERSE OF X(291) (1.0000, 0.0000) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6285, 0.2645) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5101, 0.1439) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) X_789 = o-(- 3, 0) CIRCUMCIRCLE POINT (0.1475, 0.3546) X_870 = ISOGONAL CONJUGATE OF X(869) (0.5307, 0.1650)
A
B
C
X869
X1
X76
X6
X75
X56
X85
X2
X292
X58
X274
X106
X789
X870
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) { 1, 76 }: Central Line L(1, 76) { 6, 75 }: Central Line L(6, 75) { 56, 85 }: Central Line L(56, 85) { 2, 292 }: Central Line L(2, 292) { 58, 274 }: Central Line L(58, 274) { 34, 331 }: Central Line L(34, 331)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_869 = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32) (0.6717, 0.1591) X_1 = INCENTER (0.6154, 0.1300) X_76 = 3rd BROCARD POINT (0.3495, 0.0360) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4506, 0.0607) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6494, 0.2495) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5911, 0.1796) X_2 = CENTROID (0.5444, 0.0934) X_292 = -HIRST INVERSE OF X(291) (0.9966, 0.0147) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6639, 0.1838) X_274 = ISOGONAL CONJUGATE OF X(213) (0.4902, 0.0774) X_34 = -BETH CONJUGATE OF X(4) (0.6197, 0.3018) X_331 = ISOTOMIC CONJUGATE OF X(219) (0.6714, 0.4104) X_870 = ISOGONAL CONJUGATE OF X(869) (0.5225, 0.0972)
A
B
C
X869
X1
X76
X6
X75
X56
X85
X2
X292
X58
X274
X34
X331
X870
Kimberling (6, 9, 13) | AB = 1
X870 (ISOGONAL CONJUGATE OF X869 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[a^{2}(b^{2} + c^{2} + bc)] : :\)
Search Value (6, 9, 13): \(2.762812806045\)
▸ Incident Central Lines9 lines
{1 76 2 292 6 75 34 331 56 85 58 274 86 871 106 789 767 825
▸ Functional Relations3 relations
Notes
Let A'B'C' be the trilinear obverse triangle of X2 LA be the line through A' parallel to BC , and define LB and LC cyclically. Let A″ = LB ∩LC , and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X870 Randy Hutson , November 30, 2018)
