X60
ISOGONAL CONJUGATE OF X12
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) { 36, 58 }: Central Line L(36, 58) { 28, 81 }: Central Line L(28, 81) { 1, 110 }: Central Line L(1, 110) { 86, 272 }: Central Line L(86, 272) { 283, 284 }: Central Line L(283, 284) { 21, 960 }: Central Line L(21, 960)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6285, 0.4382) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_1 = INCENTER (0.5714, 0.2333) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_272 = ISOGONAL CONJUGATE OF X(209) (-0.2458, -0.0534) X_283 = -CROSS CONJUGATE OF X(21) (0.4475, 0.0704) X_284 = -CROSS CONJUGATE OF X(21) (0.5646, 0.2088) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_960 = INTERSECTION OF LINES X(1)X(6) AND X(5)X(10) (0.5041, 0.1533) X_60 = ISOGONAL CONJUGATE OF X(12) (0.5790, 0.2258)
A
B
C
X12
X36
X58
X28
X81
X1
X110
X86
X272
X283
X284
X21
X960
X60
(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) { 36, 58 }: Central Line L(36, 58) { 28, 81 }: Central Line L(28, 81) { 1, 110 }: Central Line L(1, 110) { 86, 272 }: Central Line L(86, 272) { 283, 284 }: Central Line L(283, 284) { 21, 960 }: Central Line L(21, 960)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.0000, 1.0000) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6522, 0.2609) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6400, 0.4800) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.6185, 0.2215) X_1 = INCENTER (0.6000, 0.2000) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5711, 0.1809) X_272 = ISOGONAL CONJUGATE OF X(209) (1.9108, 0.2215) X_283 = -CROSS CONJUGATE OF X(21) (0.4286, 0.0000) X_284 = -CROSS CONJUGATE OF X(21) (0.5946, 0.1622) X_21 = SCHIFFLER POINT (0.5368, 0.1263) X_960 = INTERSECTION OF LINES X(1)X(6) AND X(5)X(10) (0.5000, 0.1000) X_60 = ISOGONAL CONJUGATE OF X(12) (0.6152, 0.1823)
A
B
C
X12
X36
X58
X28
X81
X1
X110
X86
X272
X283
X284
X21
X960
X60
(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) { 36, 58 }: Central Line L(36, 58) { 28, 81 }: Central Line L(28, 81) { 1, 110 }: Central Line L(1, 110) { 86, 272 }: Central Line L(86, 272) { 283, 284 }: Central Line L(283, 284) { 21, 960 }: Central Line L(21, 960)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.9387, 1.0546) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6285, 0.2645) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6180, 0.4859) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5997, 0.2241) X_1 = INCENTER (0.5841, 0.2021) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5599, 0.1831) X_272 = ISOGONAL CONJUGATE OF X(209) (2.1568, 0.2241) X_283 = -CROSS CONJUGATE OF X(21) (0.4401, 0.0000) X_284 = -CROSS CONJUGATE OF X(21) (0.5794, 0.1635) X_21 = SCHIFFLER POINT (0.5310, 0.1276) X_960 = INTERSECTION OF LINES X(1)X(6) AND X(5)X(10) (0.5000, 0.1010) X_60 = ISOGONAL CONJUGATE OF X(12) (0.5969, 0.1840)
A
B
C
X12
X36
X58
X28
X81
X1
X110
X86
X272
X283
X284
X21
X960
X60
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) { 36, 58 }: Central Line L(36, 58) { 28, 81 }: Central Line L(28, 81) { 1, 110 }: Central Line L(1, 110) { 86, 272 }: Central Line L(86, 272) { 283, 284 }: Central Line L(283, 284) { 21, 960 }: Central Line L(21, 960)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.7120, 0.4690) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6639, 0.1838) X_28 = CEVAPOINT OF X(19) AND X(25) (0.5818, 0.4033) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.6338, 0.1505) X_1 = INCENTER (0.6154, 0.1300) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5757, 0.1133) X_272 = ISOGONAL CONJUGATE OF X(209) (0.9317, -0.0264) X_283 = -CROSS CONJUGATE OF X(21) (0.4677, -0.0338) X_284 = -CROSS CONJUGATE OF X(21) (0.6229, 0.0691) X_21 = SCHIFFLER POINT (0.5385, 0.0448) X_960 = INTERSECTION OF LINES X(1)X(6) AND X(5)X(10) (0.4957, 0.0289) X_60 = ISOGONAL CONJUGATE OF X(12) (0.6473, 0.0852)
A
B
C
X12
X36
X58
X28
X81
X1
X110
X86
X272
X283
X284
X21
X960
X60
Kimberling (6, 9, 13) | AB = 1
X60 (ISOGONAL CONJUGATE OF X12 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[1 + \cos(B - C)] : 1/[1 + \cos(C - A)] : 1/[1 + \cos(A - B)]\)
Barycentrics: \(a/[1 + \cos(B - C)] : b/[1 + \cos(C - A)] : c/[1+ \cos(A - B)]\)
Search Value (6, 9, 13): \(1.901815483389\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines9 lines
{1 110 21 960 28 81 36 58 59 1101 86 272 283 284 404 662 757 1014
▸ Functional Relations10 relations
Notes
Let A'B'C' be the cevian triangle of X21 A″ , B″ , C″ be the inverse-in-circumcircle of A' , B' , C' . The lines AA″ , BB″ , CC″ concur in X60 Randy Hutson , October 15, 2018)
