X49
CENTER OF SINE-TRIPLE-ANGLE CIRCLE
Suppose that
P and
Q are distinct points in the plane of a triangle
ABC . Let
PA = reflection of
P in line
AQ , let
QA = reflection of
Q in line
AP , and let
MA = midpoint of segment
PA QA . Define
MB and
MC cyclically.
César Lozada found that if
Q =
isogonal conjugate of
P , then the locus of
P for which
MA MB MC is perspective to
ABC is the union of a cubic and 6 circles: specifically, the McCay cubic (K003), the circles {B,C,
Jb ,
Jc }}, {C,A,
Jc ,
Ja }}, {A,B,
Ja ,
Jb }}, and the circles {B,C,
Ja }}, {C,A,
Jb }}, {A,B,
Jc }}, where
Ja ,
Jb ,
Jc are the excenters of
ABC . Moreover, if
P =
X3 and
Q =
X4 , then
MA MB MC is not only perspective, but homothetic, to
ABC , and the center of homothety is
X49 . See
Hyacinthos 23265, June 1, 2015.
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) { 5, 54 }: Central Line L(5, 54) { 3, 155 }: Central Line L(3, 155) { 4, 156 }: Central Line L(4, 156) { 93, 94 }: Central Line L(93, 94) { 1, 215 }: Central Line L(1, 215) { 52, 195 }: Central Line L(52, 195)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_93 = ISOGONAL CONJUGATE OF X(49) (0.5299, 0.2878) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_54 = KOSNITA POINT (0.5880, 0.2176) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (1.1658, 0.5033) X_4 = ORTHOCENTER (0.6122, 0.3958) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_94 = ISOGONAL CONJUGATE OF X(50) (0.9069, -0.1331) X_1 = INCENTER (0.5714, 0.2333) X_215 = -CEVA CONJUGATE OF X(50) (0.4113, 0.3887) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_195 = -CEVA CONJUGATE OF X(3) (0.6761, 0.3331) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789)
A
B
C
X93
X5
X54
X3
X155
X4
X156
X94
X1
X215
X52
X195
X49
(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) { 5, 54 }: Central Line L(5, 54) { 3, 155 }: Central Line L(3, 155) { 4, 156 }: Central Line L(4, 156) { 93, 94 }: Central Line L(93, 94) { 1, 215 }: Central Line L(1, 215) { 52, 195 }: Central Line L(52, 195)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_93 = ISOGONAL CONJUGATE OF X(49) (0.6400, 0.4800) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_54 = KOSNITA POINT (0.6400, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (-0.0283, 0.1600) X_4 = ORTHOCENTER (0.6400, 0.4800) X_156 = -OF-TANGENTIAL-TRIANGLE (0.7077, -0.3920) X_94 = ISOGONAL CONJUGATE OF X(50) (0.8532, -0.2508) X_1 = INCENTER (0.6000, 0.2000) X_215 = -CEVA CONJUGATE OF X(50) (0.4800, 0.3333) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.5218, 0.4800) X_195 = -CEVA CONJUGATE OF X(3) (0.7800, 0.3200) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7800, 0.0000)
A
B
C
X93
X5
X54
X3
X155
X4
X156
X94
X1
X215
X52
X195
X49
(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) { 5, 54 }: Central Line L(5, 54) { 3, 155 }: Central Line L(3, 155) { 4, 156 }: Central Line L(4, 156) { 93, 94 }: Central Line L(93, 94) { 1, 215 }: Central Line L(1, 215) { 52, 195 }: Central Line L(52, 195)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_93 = ISOGONAL CONJUGATE OF X(49) (0.6180, 0.4859) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_54 = KOSNITA POINT (0.6180, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (-0.0357, 0.1620) X_4 = ORTHOCENTER (0.6180, 0.4859) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6751, -0.3968) X_94 = ISOGONAL CONJUGATE OF X(50) (0.8111, -0.3087) X_1 = INCENTER (0.5841, 0.2021) X_215 = -CEVA CONJUGATE OF X(50) (0.4809, 0.3392) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.9018, 0.4859) X_195 = -CEVA CONJUGATE OF X(3) (0.7361, 0.3239) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7361, 0.0000)
A
B
C
X93
X5
X54
X3
X155
X4
X156
X94
X1
X215
X52
X195
X49
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) { 5, 54 }: Central Line L(5, 54) { 3, 155 }: Central Line L(3, 155) { 4, 156 }: Central Line L(4, 156) { 93, 94 }: Central Line L(93, 94) { 1, 215 }: Central Line L(1, 215) { 52, 195 }: Central Line L(52, 195)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_93 = ISOGONAL CONJUGATE OF X(49) (-1.9471, 0.4523) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_54 = KOSNITA POINT (1.8800, -0.0730) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (0.3980, 0.4851) X_4 = ORTHOCENTER (0.6331, 0.8294) X_156 = -OF-TANGENTIAL-TRIANGLE (0.4965, 0.4995) X_94 = ISOGONAL CONJUGATE OF X(50) (0.6355, 0.3039) X_1 = INCENTER (0.6154, 0.1300) X_215 = -CEVA CONJUGATE OF X(50) (0.5587, 0.1842) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7045, 0.2975) X_195 = -CEVA CONJUGATE OF X(3) (3.2601, 0.1287) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.4207, 0.3163)
A
B
C
X93
X5
X54
X3
X155
X4
X156
X94
X1
X215
X52
X195
X49
Kimberling (6, 9, 13) | AB = 1
X49 (CENTER OF SINE-TRIPLE-ANGLE CIRCLE ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\cos 3A : \cos 3B : \cos 3C\)
Barycentrics: \(\sin A \cos 3A : \sin B \cos 3B : \sin C \cos 3C\)
Search Value (6, 9, 13): \(1.301655102607\)
▸ Alternate Coordinates1 form
▸ Linear Combinations1 combination
▸ Incident Central Lines8 lines
{1 215 3 155 4 156 5 54 24 568 52 195 93 94 381 578
▸ Functional Relations8 relations
Notes
V. Thebault, “Sine-triple-angle-circle,” Mathesis 65 (1956) 282-284.
