2006 APMO

[FahimFerdous]

Let A, B be two distinct points on a given circle O and let P be the midpoint of line segment AB. Let O' be the circle tangent to the line AB at P and tangent to the circle O. Let 'l' be the tangent line, different from the line AB, to O' passing through A. Let C be the intersection point, different from A, of 'l' and O. Let Q be the midpoint of the line segment BC and O" be the circle tangent to the line BC at Q and tangent to the line segment AC. Prove that the circle O" is tangent to the circle O.