Golden Section Associated to the Golden Gnomon, the Nine Point Center and an Excircle

Given:

*$ABC$ is the golden gnomon.
*$N$ is the center of the Nine-point center.
*$I_a$ is the center of the A-excircle.

Prove that the semicircle $NI_a$ touch $AC$ in $G$, such that

$$\frac{GM_b}{CG}=\phi$$