X1000
ISOGONAL CONJUGATE OF X999
End of PART 1
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) { 55, 104 }: Central Line L(55, 104) { 21, 145 }: Central Line L(21, 145) { 8, 392 }: Central Line L(8, 392) { 79, 388 }: Central Line L(79, 388) { 7, 517 }: Central Line L(7, 517) { 9, 519 }: Central Line L(9, 519)
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_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_8 = NAGEL POINT (0.4694, 0.1333) X_392 = X(1,6)∩X(10,11) (0.5265, 0.1800) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_388 = ∩X(7)X(8) (0.5790, 0.2635) X_7 = GERGONNE POINT (0.5903, 0.2769) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_9 = MITTENPUNKT (0.5110, 0.1615) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5463, 0.1961)
A
B
C
X45
X2
X55
X104
X21
X145
X8
X392
X79
X388
X7
X517
X9
X519
X1000
(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) { 55, 104 }: Central Line L(55, 104) { 21, 145 }: Central Line L(21, 145) { 8, 392 }: Central Line L(8, 392) { 79, 388 }: Central Line L(79, 388) { 7, 517 }: Central Line L(7, 517) { 9, 519 }: Central Line L(9, 519)
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_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5714, 0.1429) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_21 = SCHIFFLER POINT (0.5368, 0.1263) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.7600, 0.3200) X_8 = NAGEL POINT (0.4400, 0.0800) X_392 = X(1,6)∩X(10,11) (0.5333, 0.1333) X_79 = ISOGONAL CONJUGATE OF X(35) (0.6295, 0.3011) X_388 = ∩X(7)X(8) (0.6067, 0.2467) X_7 = GERGONNE POINT (0.6218, 0.2618) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8038, 0.6743) X_9 = MITTENPUNKT (0.5091, 0.1091) X_519 = ISOGONAL CONJUGATE OF X(106) (1.0067, 0.5050) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5673, 0.1527)
A
B
C
X45
X2
X55
X104
X21
X145
X8
X392
X79
X388
X7
X517
X9
X519
X1000
(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) { 55, 104 }: Central Line L(55, 104) { 21, 145 }: Central Line L(21, 145) { 8, 392 }: Central Line L(8, 392) { 79, 388 }: Central Line L(79, 388) { 7, 517 }: Central Line L(7, 517) { 9, 519 }: Central Line L(9, 519)
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_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5599, 0.1439) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_21 = SCHIFFLER POINT (0.5310, 0.1276) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.7182, 0.3225) X_8 = NAGEL POINT (0.4499, 0.0817) X_392 = X(1,6)∩X(10,11) (0.5280, 0.1347) X_79 = ISOGONAL CONJUGATE OF X(35) (0.6091, 0.3052) X_388 = ∩X(7)X(8) (0.5898, 0.2498) X_7 = GERGONNE POINT (0.6026, 0.2652) X_517 = ISOGONAL CONJUGATE OF X(104) (0.7602, 0.6929) X_9 = MITTENPUNKT (0.5077, 0.1103) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9672, 0.5460) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5563, 0.1539)
A
B
C
X45
X2
X55
X104
X21
X145
X8
X392
X79
X388
X7
X517
X9
X519
X1000
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) { 55, 104 }: Central Line L(55, 104) { 21, 145 }: Central Line L(21, 145) { 8, 392 }: Central Line L(8, 392) { 79, 388 }: Central Line L(79, 388) { 7, 517 }: Central Line L(7, 517) { 9, 519 }: Central Line L(9, 519)
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_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5940, 0.0549) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_21 = SCHIFFLER POINT (0.5385, 0.0448) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.8284, 0.2400) X_8 = NAGEL POINT (0.4024, 0.0200) X_392 = X(1,6)∩X(10,11) (0.5356, 0.0626) X_79 = ISOGONAL CONJUGATE OF X(35) (0.6329, 0.2757) X_388 = ∩X(7)X(8) (0.6172, 0.2016) X_7 = GERGONNE POINT (0.6288, 0.2114) X_517 = ISOGONAL CONJUGATE OF X(104) (0.7243, 0.7243) X_9 = MITTENPUNKT (0.5022, 0.0343) X_519 = ISOGONAL CONJUGATE OF X(106) (1.1273, 0.3944) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5915, 0.0805)
A
B
C
X45
X2
X55
X104
X21
X145
X8
X392
X79
X388
X7
X517
X9
X519
X1000
Kimberling (6, 9, 13) | AB = 1
X1000 (ISOGONAL CONJUGATE OF X999 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[a(a^{2} + 4bc - b^{2} - c^{2})] : :\)
Search Value (6, 9, 13): \(2.391186656011\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines11 lines
{1 631 7 517 8 392 9 519 21 145 55 104 65 3296 79 388 80 497 84 944 390 952
▸ Functional Relations4 relations
Notes
Let A* be the parabola with focus A and directrix BC , and let A* * be the polar of X1 A* . Define B* * and C* * cyclically, and let A' = B* *∩C* *, and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X1000 A'B'C' is X5657 . (Randy Hutson , July 7, 2014)
