Trilinears sin(A + π/3) : sin(B + π/3) : sin(C + π/3)
= cos(A - π/6) : cos(B - π/6) : cos(C - π/6)
Barycentrics a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)
Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)
The pedal triangle of X(15) is equilateral.
X(15) lies on these lines:
1,1251 2,14 3,6 4,17 13,30 18,140 35,1250 36,202 55,203 298,533 303,316 395,549 397,550 532,616 628,636
X(15) is the {X(3),X(6)}-harmonic conjugate of X(16).
X(15) = reflection of X(I) in X(J) for these (I,J): (13,396), (16,187), (298,618), (316,624), (621,623)
X(15) = isogonal conjugate of X(13)
X(15) = isotomic conjugate of X(300)
X(15) = inverse-in-circumcircle of X(16)
X(15) = inverse-in-Brocard-circle of X(16)
X(15) = complement of X(621)
X(15) = anticomplement of X(623)
X(15) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,202), (13,62), (74,16)
X(15) = crosspoint of X(I) and X(J) for these (I,J): (13,18), (298,470)
X(15) = crosssum of X(I) and X(J) for these (I,J): (15,62), (532,619)
X(15) = crossdifference of any two points on line X(395)X(523)
X(15) = X(6)-Hirst inverse of X(16)
, select the center name from the list and, then, click on the vertices A, B and C successively.