X1293
COLLINGS TRANSFORM OF X121
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) { 3, 106 }: Central Line L(3, 106) { 4, 121 }: Central Line L(4, 121) { 40, 104 }: Central Line L(40, 104) { 105, 165 }: Central Line L(105, 165) { 182, 727 }: Central Line L(182, 727) { 572, 739 }: Central Line L(572, 739)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3667 = com(INTOUCH TRIANGLE) (0.9481, -0.2192) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_4 = ORTHOCENTER (0.6122, 0.3958) X_121 = -OF-MEDIAL-TRIANGLE (0.3061, 0.2999) X_40 = BEVAN POINT (0.4286, -0.0292) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (1.0060, 0.1685) X_572 = ORTHOGONAL OF X(58) (0.5253, 0.1439) X_739 = EVEN (- 1, 0) CIRCUMCIRCLE POINT (1.0101, 0.0884) X_1293 = COLLINGS TRANSFORM OF X(121) (0.0000, 0.2041)
A
B
C
X3667
X3
X106
X4
X121
X40
X104
X105
X165
X182
X727
X572
X739
X1293
(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) { 3, 106 }: Central Line L(3, 106) { 4, 121 }: Central Line L(4, 121) { 40, 104 }: Central Line L(40, 104) { 105, 165 }: Central Line L(105, 165) { 182, 727 }: Central Line L(182, 727) { 572, 739 }: Central Line L(572, 739)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3667 = com(INTOUCH TRIANGLE) (0.8917, -0.3000) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_4 = ORTHOCENTER (0.6400, 0.4800) X_121 = -OF-MEDIAL-TRIANGLE (0.3200, 0.2400) X_40 = BEVAN POINT (0.4000, -0.2000) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9800, -0.1400) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4667, -0.0667) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5700, 0.1200) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (0.9576, 0.2016) X_572 = ORTHOGONAL OF X(58) (0.5361, 0.0619) X_739 = EVEN (- 1, 0) CIRCUMCIRCLE POINT (0.9869, 0.1139) X_1293 = COLLINGS TRANSFORM OF X(121) (0.0000, 0.0000)
A
B
C
X3667
X3
X106
X4
X121
X40
X104
X105
X165
X182
X727
X572
X739
X1293
(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) { 3, 106 }: Central Line L(3, 106) { 4, 121 }: Central Line L(4, 121) { 40, 104 }: Central Line L(40, 104) { 105, 165 }: Central Line L(105, 165) { 182, 727 }: Central Line L(182, 727) { 572, 739 }: Central Line L(572, 739)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3667 = com(INTOUCH TRIANGLE) (0.9234, -0.2659) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_106 = Λ(INCENTER, CENTROID) (0.9957, -0.0651) X_4 = ORTHOCENTER (0.6180, 0.4859) X_121 = -OF-MEDIAL-TRIANGLE (0.3111, 0.2755) X_40 = BEVAN POINT (0.4159, -0.2021) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9563, -0.2045) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4720, -0.0674) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5590, 0.1215) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (0.9725, 0.1635) X_572 = ORTHOGONAL OF X(58) (0.5303, 0.0625) X_739 = EVEN (- 1, 0) CIRCUMCIRCLE POINT (0.9964, 0.0601) X_1293 = COLLINGS TRANSFORM OF X(121) (0.0043, 0.0651)
A
B
C
X3667
X3
X106
X4
X121
X40
X104
X105
X165
X182
X727
X572
X739
X1293
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) { 3, 106 }: Central Line L(3, 106) { 4, 121 }: Central Line L(4, 121) { 40, 104 }: Central Line L(40, 104) { 105, 165 }: Central Line L(105, 165) { 182, 727 }: Central Line L(182, 727) { 572, 739 }: Central Line L(572, 739)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3667 = com(INTOUCH TRIANGLE) (0.8454, -0.4895) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_106 = Λ(INCENTER, CENTROID) (1.0294, -0.0621) X_4 = ORTHOCENTER (0.6331, 0.8294) X_121 = -OF-MEDIAL-TRIANGLE (0.3019, 0.1711) X_40 = BEVAN POINT (0.3846, -0.6794) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (1.0692, -0.3135) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4615, -0.4096) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5787, -0.0546) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (0.8744, 0.1558) X_572 = ORTHOGONAL OF X(58) (0.5421, -0.1570) X_739 = EVEN (- 1, 0) CIRCUMCIRCLE POINT (0.9485, 0.0778) X_1293 = COLLINGS TRANSFORM OF X(121) (-0.0294, -0.4872)
A
B
C
X3667
X3
X106
X4
X121
X40
X104
X105
X165
X182
X727
X572
X739
X1293
Kimberling (6, 9, 13) | AB = 1
X1293 (COLLINGS TRANSFORM OF X121 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where \(f(a,b,c) = a/[(b - c)(3a - b - c)]\) (M. Iliev, 5/13/07)
Search Value (6, 9, 13): \(13.154577400069\)
X1293
▸ Incident Central Lines6 lines
{3 106 4 121 40 104 105 165 182 727 572 739
▸ Functional Relations14 relations
Notes
Let LA be the reflection of the line X1 X2 BC , and define LB and LC cyclically. Let A' = LB ∩LC , and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X1293 Randy Hutson , 9/23/2011)
