A Right Triangle Rotating About a Vertex

In the coordinate plane, the point $A$ lies in the first quadrant (Fig. 1). The foot of the perpendicular from $A$ to the $x$-axis is $B$, the foot of the perpendicular from $A$ to the $y$-axis is $C$, and $AC=3$. The point $M$ lies on segment $OB$, and $\angle BCM=\angle BCA=30^\circ$.

1. Find the coordinates of the points $A$ and $M$.

2. As in Fig. 2, draw the line $MN$ through $M$ parallel to $BC$. A point $E$ moves along segment $BC$, and $F$ is the foot of the perpendicular from $E$ to line $MN$. Find the minimum value of $AE+EF+FC$.

3. As in Fig. 3, rotate $\triangle ABC$ clockwise about the point $C$, and denote the rotated triangle by $\triangle A'B'C'$ (so $C'=C$). Suppose that during the rotation the rays $CA'$ and $CB'$ both meet the $x$-axis, at points $G$ and $H$ respectively. Can $\triangle GBC$ be isosceles? If so, determine all corresponding values of $\dfrac{HG}{HC}$; if not, explain why.