Triangle inscribed in a circle and with sides going through three given points B, C, D (Castillon's problem)

1) Take three points b, c, d in the circle.
2) The line bD cuts the circle at b1.
3) The line b1B cuts the circle at b2.
4) The line b2C cuts the circle at b'.
5) Repeat the steps  2, 3, 4 with c instead b, and get c'; and with d instead b, and get d'.
6) The lines bc', cb' intersect at M.
7) The lines bd', b'd intersect at N.
8) The line MN cuts the circle in a vertex X of the requested triangle.