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Prove that point lies on the incircle.

Source: Sharygin contest 2008. The correspondence round. Problem 13

September 3, 2008
geometryincentergeometry proposed

Problem Statement

(A.Myakishev, 9--10) Given triangle ABC ABC. One of its excircles is tangent to the side BC BC at point A1 A_1 and to the extensions of two other sides. Another excircle is tangent to side AC AC at point B1 B_1. Segments AA1 AA_1 and BB1 BB_1 meet at point N N. Point P P is chosen on the ray AA1 AA_1 so that AP\equal{}NA_1. Prove that P P lies on the incircle.
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