X2477
EXSIMILICENTER{INCIRCLE, SINE-TRIPLE-ANGLE CIRCLE}
X2477 = perspector of
ABC and extraversion triangle of
X215
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) { 1, 49 }: Central Line L(1, 49) { 11, 54 }: Central Line L(11, 54) { 12, 110 }: Central Line L(12, 110) { 56, 184 }: Central Line L(56, 184) { 55, 1069 }: Central Line L(55, 1069) { 156, 1478 }: Central Line L(156, 1478)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_54 = KOSNITA POINT (0.5880, 0.2176) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_1069 = -CROSS CONJUGATE OF X(3) (0.0709, 0.0059) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_1478 = CENTER OF JOHNSON-YFF CIRCLE (0.5842, 0.2843) X_2477 = EXSIMILICENTER{INCIRCLE, SINE-TRIPLE-ANGLE CIRCLE} (0.5953, 0.2101)
A
B
C
X1
X49
X11
X54
X12
X110
X56
X184
X55
X1069
X156
X1478
X2477
(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) { 1, 49 }: Central Line L(1, 49) { 11, 54 }: Central Line L(11, 54) { 12, 110 }: Central Line L(12, 110) { 56, 184 }: Central Line L(56, 184) { 55, 1069 }: Central Line L(55, 1069) { 156, 1478 }: Central Line L(156, 1478)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1 = INCENTER (0.6000, 0.2000) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7800, 0.0000) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_54 = KOSNITA POINT (0.6400, 0.1600) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6667, 0.3333) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.6400, 0.0000) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5714, 0.1429) X_1069 = -CROSS CONJUGATE OF X(3) (-0.0283, 0.1600) X_156 = -OF-TANGENTIAL-TRIANGLE (0.7077, -0.3920) X_1478 = CENTER OF JOHNSON-YFF CIRCLE (0.6114, 0.2800) X_2477 = EXSIMILICENTER{INCIRCLE, SINE-TRIPLE-ANGLE CIRCLE} (0.6514, 0.1429)
A
B
C
X1
X49
X11
X54
X12
X110
X56
X184
X55
X1069
X156
X1478
X2477
(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) { 1, 49 }: Central Line L(1, 49) { 11, 54 }: Central Line L(11, 54) { 12, 110 }: Central Line L(12, 110) { 56, 184 }: Central Line L(56, 184) { 55, 1069 }: Central Line L(55, 1069) { 156, 1478 }: Central Line L(156, 1478)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1 = INCENTER (0.5841, 0.2021) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7361, 0.0000) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_54 = KOSNITA POINT (0.6180, 0.1620) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6411, 0.3392) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.6180, 0.0000) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5599, 0.1439) X_1069 = -CROSS CONJUGATE OF X(3) (-0.0357, 0.1620) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6751, -0.3968) X_1478 = CENTER OF JOHNSON-YFF CIRCLE (0.5938, 0.2838) X_2477 = EXSIMILICENTER{INCIRCLE, SINE-TRIPLE-ANGLE CIRCLE} (0.6278, 0.1439)
A
B
C
X1
X49
X11
X54
X12
X110
X56
X184
X55
X1069
X156
X1478
X2477
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) { 1, 49 }: Central Line L(1, 49) { 11, 54 }: Central Line L(11, 54) { 12, 110 }: Central Line L(12, 110) { 56, 184 }: Central Line L(56, 184) { 55, 1069 }: Central Line L(55, 1069) { 156, 1478 }: Central Line L(156, 1478)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1 = INCENTER (0.6154, 0.1300) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.4207, 0.3163) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_54 = KOSNITA POINT (1.8800, -0.0730) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6494, 0.2495) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.3215, 1.0549) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5940, 0.0549) X_1069 = -CROSS CONJUGATE OF X(3) (0.6796, 0.0252) X_156 = -OF-TANGENTIAL-TRIANGLE (0.4965, 0.4995) X_1478 = CENTER OF JOHNSON-YFF CIRCLE (0.6187, 0.2598) X_2477 = EXSIMILICENTER{INCIRCLE, SINE-TRIPLE-ANGLE CIRCLE} (0.7509, 0.0004)
A
B
C
X1
X49
X11
X54
X12
X110
X56
X184
X55
X1069
X156
X1478
X2477
Kimberling (6, 9, 13) | AB = 1
X2477 (EXSIMILICENTER{INCIRCLE, SINE-TRIPLE-ANGLE CIRCLE} ) 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 \(f(a,b,c) = a^{3}(a^{2} - b^{2} - c^{2} - bc)^{2}/(b + c - a)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.960686846995\)
▸ Alternate Coordinates2 forms
Trilinears: \(\sin^{2}(3A/2) : \sin^{2}(3B/2) : \sin^{2}(3C/2)\) (M. Iliev, 4/12/07)
Trilinears: \(1 - \cos 3A : 1 - \cos 3B : 1 - \cos 3C\) (M. Iliev, 4/12/07)
▸ Incident Central Lines7 lines
{1 49 11 54 12 110 55 1069 56 184 156 1478 2595 2606
▸ Functional Relations1 relation
