X3602
ISOGONAL CONJUGATE OF X3274
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) { 356, 357 }: Central Line L(356, 357) { 1134, 1137 }: Central Line L(1134, 1137) { 1135, 1136 }: Central Line L(1135, 1136) { 3274, 3604 }: Central Line L(3274, 3604) { 3275, 3603 }: Central Line L(3275, 3603)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3274 = PERSPECTOR OF TRIANGLES P1P2P3 AND J1J2J3 (0.5554, 0.2055) X_356 = MORLEY CENTER (0.5727, 0.2361) X_357 = 1st MORLEY-TAYLOR-MARR CENTER (0.5757, 0.2422) X_1134 = 3rd MORLEY-TAYLOR-MARR CENTER (0.5584, 0.2154) X_1137 = 6th MORLEY-TAYLOR-MARR CENTER (0.4890, 0.0964) X_1135 = 4th MORLEY-TAYLOR-MARR CENTER (0.5834, 0.2515) X_1136 = 5th MORLEY-TAYLOR-MARR CENTER (0.6158, 0.4039) X_3604 = ISOGONAL CONJUGATE OF X(3273) (0.5684, 0.2299) X_3275 = PERSPECTOR OF TRIANGLES S1S2S3 AND J1J2J3 (0.6086, 0.2928) X_3603 = ISOGONAL CONJUGATE OF X(3275) (0.5222, 0.1769) X_3602 = ISOGONAL CONJUGATE OF X(3274) (0.5856, 0.2621)
A
B
C
X3274
X356
X357
X1134
X1137
X1135
X1136
X3604
X3275
X3603
X3602
(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) { 356, 357 }: Central Line L(356, 357) { 1134, 1137 }: Central Line L(1134, 1137) { 1135, 1136 }: Central Line L(1135, 1136) { 3274, 3604 }: Central Line L(3274, 3604) { 3275, 3603 }: Central Line L(3275, 3603)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_3274 = PERSPECTOR OF TRIANGLES P1P2P3 AND J1J2J3 (0.5791, 0.1619) X_356 = MORLEY CENTER (0.6015, 0.2039) X_357 = 1st MORLEY-TAYLOR-MARR CENTER (0.6052, 0.2125) X_1134 = 3rd MORLEY-TAYLOR-MARR CENTER (0.5820, 0.1772) X_1137 = 6th MORLEY-TAYLOR-MARR CENTER (0.4827, 0.0000) X_1135 = 4th MORLEY-TAYLOR-MARR CENTER (0.6156, 0.2233) X_1136 = 5th MORLEY-TAYLOR-MARR CENTER (0.6400, 0.4800) X_3604 = ISOGONAL CONJUGATE OF X(3273) (0.5958, 0.1960) X_3275 = PERSPECTOR OF TRIANGLES S1S2S3 AND J1J2J3 (0.6467, 0.2755) X_3603 = ISOGONAL CONJUGATE OF X(3275) (0.5249, 0.1289) X_3602 = ISOGONAL CONJUGATE OF X(3274) (0.6172, 0.2400)
A
B
C
X3274
X356
X357
X1134
X1137
X1135
X1136
X3604
X3275
X3603
X3602
(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) { 356, 357 }: Central Line L(356, 357) { 1134, 1137 }: Central Line L(1134, 1137) { 1135, 1136 }: Central Line L(1135, 1136) { 3274, 3604 }: Central Line L(3274, 3604) { 3275, 3603 }: Central Line L(3275, 3603)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_3274 = PERSPECTOR OF TRIANGLES P1P2P3 AND J1J2J3 (0.5664, 0.1633) X_356 = MORLEY CENTER (0.5853, 0.2061) X_357 = 1st MORLEY-TAYLOR-MARR CENTER (0.5885, 0.2148) X_1134 = 3rd MORLEY-TAYLOR-MARR CENTER (0.5689, 0.1789) X_1137 = 6th MORLEY-TAYLOR-MARR CENTER (0.4854, 0.0000) X_1135 = 4th MORLEY-TAYLOR-MARR CENTER (0.5973, 0.2259) X_1136 = 5th MORLEY-TAYLOR-MARR CENTER (0.6180, 0.4859) X_3604 = ISOGONAL CONJUGATE OF X(3273) (0.5805, 0.1981) X_3275 = PERSPECTOR OF TRIANGLES S1S2S3 AND J1J2J3 (0.6238, 0.2794) X_3603 = ISOGONAL CONJUGATE OF X(3275) (0.5210, 0.1304) X_3602 = ISOGONAL CONJUGATE OF X(3274) (0.5987, 0.2429)
A
B
C
X3274
X356
X357
X1134
X1137
X1135
X1136
X3604
X3275
X3603
X3602
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) { 356, 357 }: Central Line L(356, 357) { 1134, 1137 }: Central Line L(1134, 1137) { 1135, 1136 }: Central Line L(1135, 1136) { 3274, 3604 }: Central Line L(3274, 3604) { 3275, 3603 }: Central Line L(3275, 3603)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_3274 = PERSPECTOR OF TRIANGLES P1P2P3 AND J1J2J3 (0.5980, 0.0798) X_356 = MORLEY CENTER (0.6164, 0.1355) X_357 = 1st MORLEY-TAYLOR-MARR CENTER (0.6190, 0.1470) X_1134 = 3rd MORLEY-TAYLOR-MARR CENTER (0.5984, 0.1051) X_1137 = 6th MORLEY-TAYLOR-MARR CENTER (0.4794, -0.2168) X_1135 = 4th MORLEY-TAYLOR-MARR CENTER (0.6282, 0.1555) X_1136 = 5th MORLEY-TAYLOR-MARR CENTER (0.6073, 0.6791) X_3604 = ISOGONAL CONJUGATE OF X(3273) (0.6113, 0.1269) X_3275 = PERSPECTOR OF TRIANGLES S1S2S3 AND J1J2J3 (0.6500, 0.2099) X_3603 = ISOGONAL CONJUGATE OF X(3275) (0.5181, 0.0541) X_3602 = ISOGONAL CONJUGATE OF X(3274) (0.6271, 0.1829)
A
B
C
X3274
X356
X357
X1134
X1137
X1135
X1136
X3604
X3275
X3603
X3602
Kimberling (6, 9, 13) | AB = 1
X3602 (ISOGONAL CONJUGATE OF X3274 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\csc(A/3 - \pi /3) : \csc(B/3 - \pi /3) : \csc(C/3 - \pi /3)\)
Barycentrics: \((\sin A)\csc(A/3 - \pi /3) : (\sin B)\csc(B/3 - \pi /3) : (\sin C)\csc(C/3 - \pi /3)\)
Search Value (6, 9, 13): \(1.051343278177\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines5 lines
▸ Functional Relations2 relations
Notes
Let A'B'C' be the 1st Morley triangle. Let A″ be the trilinear pole of line B'C' , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X3602 Randy Hutson , July 11, 2019)
Let A'B'C' be the 1st Morley triangle. Let A* be the foot of the perpendicular from A to line B'C' , and define B* and C* cyclically. The lines A'A* , B'B* , C'C* concur in X3602
Let A'B'C' be the 2nd Morley triangle. Let Ba and Ca be points on BC such that A'Bc Ca is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A″ be the midpoint of Ab and Ac , and define B″ and C″ cyclically. Then A″B″C″ , here named the 2nd MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC , with perspector X3602 Thanh Oai Dao and Peter Moses , March 30, 2018)
