The tan-colored rotating triangle a(t)b(t)c(t) has the same orientation as triangle ABC. Lines B-to-b(t) and C-to-c(t) meet in a point u, and points v and w are defined cyclically. The orbits of u, v, and w concur in a point. Thus, as the triangle rotates, there is only one position in which the lines A-to-a(t), B-to-b(t), and C-to-c(t) concur.
For more, see GEOMETRY IN ACTION, Chapter 7.