X(6149)=X(13)-Isoconjugate of X(14), (the last center published in ETC by Dec 12, 2014) has the very nice barycentrics coordinates F(A) : F(B) : F(C), where F(A) = sin(3*A).
The next table shows other centers with similar simple and nice trigonometric coordinates, when these represent trilinears or barycentrics and when points can be found from other known centers.
F(A) | Center with trilinears F(A):F(B):F(C) | Center with barycentrics F(A):F(B):F(C) |
sin(A) | X(6) | X(1) |
sin(2*A) | X(48) | X(3) |
sin(3*A) | X(50) | X(6149) |
sin(4*A) | X(563) | X(1147) |
sin(6*A) | (47,48)∩(50,2477) | Complement of X(562) (2,562)∩(3,54) |
sin(2*A)^2 | (1,1748)∩(31,48) | X(577) |
sin(2*A)^3 | (3,54)∩(4,2055) | |
sin(2*A)^4 | (577,1147)∩(1971,2055) | |
sin(3*A)^2 | (50,215)∩(1109,2619) | (6,1511)∩(1971,3258) |
sin(3*A)^3 | (49,50)∩(54,2088) | |
cos(A) | X(3) | X(63) |
cos(2*A) | X(47) | X(1993) |
cos(3*A) | X(49) | (48,63)∩(662,2167) |
cos(4*A) | Eigencenter of anticevian triangle of X(563) On line (1,1748) | (2,95)∩(50,1993) |
cos(A)^2 | X(255) | X(394) |
cos(A)^3 | X(1092) | (48,63)∩(92,1958) |
cos(A)^4 | (1,775)∩(47,560) | (2,801)∩(32,1993) |
cos(2*A)^2 | (1,1748)∩(560,2964) | (2,95)∩(32,1994) |
tan(A) | X(19) | X(4) |
tan(2*A) | X(1820) | X(68) |
tan(3*A) | X(562) | |
tan(A)^2 | X(1096) | X(393) |
tan(A)^3 | Isogonal conjugate of X(1102) (19,158)∩(811,2128) | Isogonal conjugate of X(3964) (4,51)∩(25,393) |
tan(A)^4 | Polar conjugate of X(4176) (133,3863)∩(185,1208) | |
tan(2*A)^2 | (68,577)∩(216,2165) |
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