Problem

Source: China Western Mathematical Olympiad 2008

Tags: geometry, function, symmetry, ratio, geometry proposed



In triangle $ ABC$, $ AB=AC$, the inscribed circle $ I$ touches $ BC, CA, AB$ at points $ D,E$ and $ F$ respectively. $ P$ is a point on arc $ EF$ opposite $ D$. Line $ BP$ intersects circle $ I$ at another point $ Q$, lines $ EP$, $ EQ$ meet line $ BC$ at $ M, N$ respectively. Prove that (1) $ P, F, B, M$ concyclic (2)$ \frac{EM}{EN} = \frac{BD}{BP}$ (P.S. Can anyone help me with using GeoGebra, the incircle function of the plugin doesn't work with my computer.)