Sharygin 2008, correspondence round, problem 13
Sharygin 2008, correspondence round, problem 13 is a plane geometry problem from the Sharygin Geometry Olympiad 2008, correspondence round, problem 13 (grades 9–10): excircle touch points and a point on the incircle.
Problem
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.
Source Sharygin Geometry Olympiad 2008, correspondence round, problem 13 (grades 9–10) · proposed by A. Myakishev
The diagram
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