X1292
COLLINGS TRANSFORM OF X55
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, 105 }: Central Line L(3, 105) { 4, 120 }: Central Line L(4, 120) { 40, 103 }: Central Line L(40, 103) { 104, 376 }: Central Line L(104, 376) { 378, 915 }: Central Line L(378, 915) { 601, 727 }: Central Line L(601, 727)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3309 = ISOGONAL CONJUGATE OF X(1292) (0.0885, 0.5779) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_4 = ORTHOCENTER (0.6122, 0.3958) X_120 = -OF-MEDIAL-TRIANGLE (0.3204, 0.3466) X_40 = BEVAN POINT (0.4286, -0.0292) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_378 = REFLECTION OF X(22) IN X(3) (0.5857, 0.3262) X_915 = ISOGONAL CONJUGATE OF X(912) (0.9488, 0.3450) X_601 = INTERSECTION OF LINES X(3)X(31) AND X(55)X(255) (0.5761, 0.2252) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (1.0060, 0.1685) X_1292 = COLLINGS TRANSFORM OF X(55) (0.0286, 0.2974)
A
B
C
X3309
X3
X105
X4
X120
X40
X103
X104
X376
X378
X915
X601
X727
X1292
(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, 105 }: Central Line L(3, 105) { 4, 120 }: Central Line L(4, 120) { 40, 103 }: Central Line L(40, 103) { 104, 376 }: Central Line L(104, 376) { 601, 727 }: Central Line L(601, 727) { 517, 840 }: Central Line L(517, 840)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3309 = ISOGONAL CONJUGATE OF X(1292) (0.1401, 0.5666) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9800, -0.1400) X_4 = ORTHOCENTER (0.6400, 0.4800) X_120 = -OF-MEDIAL-TRIANGLE (0.3300, 0.3100) X_40 = BEVAN POINT (0.4000, -0.2000) X_103 = ANTIPODE OF X(101) (0.6923, -0.4615) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4533, -0.1600) X_601 = INTERSECTION OF LINES X(3)X(31) AND X(55)X(255) (0.6071, 0.1786) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (0.9576, 0.2016) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8038, 0.6743) X_840 = ISOGONAL CONJUGATE OF X(528) (0.1000, 0.3000) X_1292 = COLLINGS TRANSFORM OF X(55) (0.0200, 0.1400)
A
B
C
X3309
X3
X105
X4
X120
X40
X103
X104
X376
X601
X727
X517
X840
X1292
(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, 105 }: Central Line L(3, 105) { 4, 120 }: Central Line L(4, 120) { 40, 103 }: Central Line L(40, 103) { 104, 376 }: Central Line L(104, 376) { 601, 727 }: Central Line L(601, 727) { 517, 840 }: Central Line L(517, 840)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3309 = ISOGONAL CONJUGATE OF X(1292) (0.0973, 0.5297) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9563, -0.2045) X_4 = ORTHOCENTER (0.6180, 0.4859) X_120 = -OF-MEDIAL-TRIANGLE (0.3309, 0.3452) X_40 = BEVAN POINT (0.4159, -0.2021) X_103 = ANTIPODE OF X(101) (0.6635, -0.4725) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4607, -0.1620) X_601 = INTERSECTION OF LINES X(3)X(31) AND X(55)X(255) (0.5902, 0.1804) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (0.9725, 0.1635) X_517 = ISOGONAL CONJUGATE OF X(104) (0.7602, 0.6929) X_840 = ISOGONAL CONJUGATE OF X(528) (0.0333, 0.1794) X_1292 = COLLINGS TRANSFORM OF X(55) (0.0437, 0.2045)
A
B
C
X3309
X3
X105
X4
X120
X40
X103
X104
X376
X601
X727
X517
X840
X1292
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, 105 }: Central Line L(3, 105) { 4, 120 }: Central Line L(4, 120) { 40, 103 }: Central Line L(40, 103) { 104, 376 }: Central Line L(104, 376) { 378, 915 }: Central Line L(378, 915) { 601, 727 }: Central Line L(601, 727)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3309 = ISOGONAL CONJUGATE OF X(1292) (0.1209, 0.5944) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (1.0692, -0.3135) X_4 = ORTHOCENTER (0.6331, 0.8294) X_120 = -OF-MEDIAL-TRIANGLE (0.2820, 0.2968) X_40 = BEVAN POINT (0.3846, -0.6794) X_103 = ANTIPODE OF X(101) (0.5518, -0.8428) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4556, -0.6427) X_378 = REFLECTION OF X(22) IN X(3) (0.1672, -3.0347) X_915 = ISOGONAL CONJUGATE OF X(912) (-0.0546, -0.4084) X_601 = INTERSECTION OF LINES X(3)X(31) AND X(55)X(255) (0.6208, 0.0505) X_727 = EVEN (- 2, -1) CIRCUMCIRCLE POINT (0.8744, 0.1558) X_1292 = COLLINGS TRANSFORM OF X(55) (-0.0692, -0.2359)
A
B
C
X3309
X3
X105
X4
X120
X40
X103
X104
X376
X378
X915
X601
X727
X1292
Kimberling (6, 9, 13) | AB = 1
X1292 (COLLINGS TRANSFORM OF X55 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/((b - c) (a^{2} + b^{2} + c^{2} - 2 a b - 2 a c)) : :\)
Search Value (6, 9, 13): \(10.870796601446\)
X1292
▸ Incident Central Lines9 lines
{3 105 4 120 40 103 104 376 378 915 411 1311 517 840 601 727 906 919
▸ Functional Relations14 relations
Notes
Let LA be the reflection of the line X1 X6 BC , and define LB and LC cyclically. Let A' = LB ∩LC , and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X1292 MA be the reflection of the line X7 X8 BC , and define MB and MC cyclically. Let A″ = MB ∩MC , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X1292 Randy Hutson , 9/23/2011)
Let Ja Jb Jc be the excentral triangle . The van Aubel lines of triangles Ja BC , Jb CA , Jc AB bound a triangle perspective to ABC at X1292
