Question 149
Let $ABC$ be an acute triangle and let $D$ be the foot of the bisector from $A$. Let $\omega$ be the circle through $A$ tangent to $BC$ at $D$, and let $E, F$ be the intersections of $\omega$ with $AB, AC$ respectively. The tangents to $\omega$ at $E$ and $F$ meet at $P$. Given that $PE=3$ and that the radius of $\omega$ is $4$, what is the length of segment $PD$?
(A) $\frac{3}{4}$
(B) $1$
(C) $\frac{1+\sqrt{2}}{2}$
(D) $\sqrt{2}$
(E) The data are not sufficient to determine it
(A) $\frac{3}{4}$
(B) $1$
(C) $\frac{1+\sqrt{2}}{2}$
(D) $\sqrt{2}$
(E) The data are not sufficient to determine it