HYACINTHOS 8074

If A'B'C' is the Yff central triangle of a
triangle ABC, then A' has trilinear coordinates

( cos(B/2) cos(C/2)
: cos(C/2)(cos(C/2) + cos(A/2))
: cos(B/2)(cos(A/2) + cos(B/2)) ).

On the other hand, let I be the center of the
incircle of triangle ABC, and X, Y, Z the first
intersections of the lines AI, BI, CI with the
incircle. The tangents to the incircle at X, Y,
Z enclose a triangle X'Y'Z'. This triangle is
called "tangential mid-arc triangle" of triangle
ABC. The vertex X' has trilinears

( - cos(B/2) cos(C/2)
: cos(C/2)(cos(C/2) + cos(A/2))
: cos(B/2)(cos(A/2) + cos(B/2)) ).

Hence, the point X' is the A-triharmonic
conjugate of A' with respect to triangle ABC.

[The A-triharmonic conjugate of a point P is
the A-vertex of the anticevian triangle of P.]

I wonder if this was known.

Darij Grinberg