Question 140
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 $\widehat{EF}$ (not containing $D$). Line $BP$ intersects the circle $I$ at another point $Q$, and lines $EP$, $EQ$ meet line $BC$ at $M$, $N$ respectively. Prove that
(1) $P$, $F$, $B$, $M$ are concyclic;
$$
(2) \frac{EM}{EN} = \frac{BD}{BP}.
$$
(1) $P$, $F$, $B$, $M$ are concyclic;
$$
(2) \frac{EM}{EN} = \frac{BD}{BP}.
$$