X89
ISOGONAL CONJUGATE OF X45
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, 44 }: Central Line L(2, 44) { 6, 88 }: Central Line L(6, 88) { 1, 902 }: Central Line L(1, 902) { 649, 1022 }: Central Line L(649, 1022)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_45 = -BETH CONJUGATE OF X(1) (0.5418, 0.1981) X_2 = CENTROID (0.5374, 0.2000) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_1 = INCENTER (0.5714, 0.2333) X_902 = CROSSPOINT OF X(6) AND X(106) (1.2500, 1.2247) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-21.0000, 14.6969) X_1022 = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514) (1.0000, -0.0000) X_89 = ISOGONAL CONJUGATE OF X(45) (0.5963, 0.2697)
A
B
C
X45
X2
X44
X6
X88
X1
X902
X649
X1022
X89
(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, 44 }: Central Line L(2, 44) { 6, 88 }: Central Line L(6, 88) { 1, 902 }: Central Line L(1, 902) { 649, 1022 }: Central Line L(649, 1022)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_45 = -BETH CONJUGATE OF X(1) (0.5565, 0.1565) X_2 = CENTROID (0.5467, 0.1600) X_44 = -LINE CONJUGATE OF X(1) (1.6000, 1.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_1 = INCENTER (0.6000, 0.2000) X_902 = CROSSPOINT OF X(6) AND X(106) (1.0000, 0.7500) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-8.0000, 6.0000) X_1022 = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514) (1.0000, -0.0000) X_89 = ISOGONAL CONJUGATE OF X(45) (0.6327, 0.2449)
A
B
C
X45
X2
X44
X6
X88
X1
X902
X649
X1022
X89
(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, 44 }: Central Line L(2, 44) { 6, 88 }: Central Line L(6, 88) { 1, 902 }: Central Line L(1, 902) { 649, 1022 }: Central Line L(649, 1022)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_45 = -BETH CONJUGATE OF X(1) (0.5474, 0.1580) X_2 = CENTROID (0.5393, 0.1620) X_44 = -LINE CONJUGATE OF X(1) (1.5111, 1.3165) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9710, -0.0490) X_1 = INCENTER (0.5841, 0.2021) X_902 = CROSSPOINT OF X(6) AND X(106) (0.9511, 0.8099) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-9.6244, 5.9482) X_1022 = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514) (1.0200, 0.0207) X_89 = ISOGONAL CONJUGATE OF X(45) (0.6118, 0.2481)
A
B
C
X45
X2
X44
X6
X88
X1
X902
X649
X1022
X89
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, 44 }: Central Line L(2, 44) { 6, 88 }: Central Line L(6, 88) { 1, 902 }: Central Line L(1, 902) { 649, 1022 }: Central Line L(649, 1022)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_45 = -BETH CONJUGATE OF X(1) (0.5647, 0.0872) X_2 = CENTROID (0.5444, 0.0934) X_44 = -LINE CONJUGATE OF X(1) (1.1019, 0.5412) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_88 = ISOGONAL CONJUGATE OF X(44) (0.9802, -0.0393) X_1 = INCENTER (0.6154, 0.1300) X_902 = CROSSPOINT OF X(6) AND X(106) (0.7729, 0.3671) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-2.9286, 1.6903) X_1022 = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514) (1.0171, 0.0155) X_89 = ISOGONAL CONJUGATE OF X(45) (0.6444, 0.1737)
A
B
C
X45
X2
X44
X6
X88
X1
X902
X649
X1022
X89
Kimberling (6, 9, 13) | AB = 1
X89 (ISOGONAL CONJUGATE OF X45 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/(2b + 2c - a) : 1/(2c + 2a - b) : 1/(2a + 2b - c)\)
Barycentrics: \(a/(2b + 2c - a) : b/(2c + 2a - b) : c/(2a + 2b - c)\)
Search Value (6, 9, 13): \(1.010174849101\)
▸ Incident Central Lines4 lines
{1 902 2 44 6 88 649 1022
▸ Functional Relations5 relations
Notes
Let A9 B9 C9 be the Gemini triangle 9. Let LA be the line through A9 parallel to BC , and define LB and LC cyclically. Let A'9 = LB ∩LC , and define B'9 , C'9 cyclically. Triangle A'9 B'9 C'9 is homothetic to ABC at X89 Randy Hutson , November 30, 2018)
