KZ-MO 2006 Problem 4

grade IX P4, X P3 The bisectors of the angles $A$ and $C$ of the triangle $ABC$ intersect the circumscirbed circle of this triangle at the points $A_0$ and $C_0$ , respectively. The straight line passing through the center of the inscribed circle of triangle $ABC$ parallel to the side of $AC$ , intersects with the line $A_0C_0$ at $P$ . Prove that the line $PB$ is tangent to the circumcircle of the triangle $ABC$ .

grade XI P4 The bisectors of the angles $A$ and $C$ of the triangle $ABC$ intersect the sides at the points $A_1$ and $C_1$ , and the circumcircle of this triangle at points $A_0$ and $C_0$ respectively. Straight lines $A_1C_1$ and $A_0C_0$ intersect at point $P$ . Prove that the segment connecting $P$ with the center inscribed circles of triangle $ABC$ , parallel to $AC$ .