SHARYGIN 2008 Problem 13

(A.Myakishev, 9--10) Given triangle $ABC$ . One of its excircles is tangent to the side $BC$ at point $A_1$ and to the extensions of two other sides. Another excircle is tangent to side $AC$ at point $B_1$ . Segments $AA_1$ and $BB_1$ meet at point $N$ . Point $P$ is chosen on the ray $AA_1$ so that $AP=NA_1$ . Prove that $P$ lies on the incircle.