What is the angle $\theta$, formed between two radii of length $r$ extending from a circle's center to its circumference, that provides the maximum ratio between the area of another circle (of maximum area) inscribed in the isosceles triangle (formed by two these two radii and the chord connecting them), and the area of the shaded region depicted in the link (namely the difference between the triangle's area and that of the circle it contains)?
More simply, what is the angle $\theta$ that gives the maximum ratio of the small circle to the shaded region?
In $\triangle OMA$, $AM = R \sin \phi$.
In $\triangle IMA$, $r = (R \sin \phi ) (\tan (\dfrac {\pi}{4} -\dfrac {\phi}{2})) = … = (R \sin \phi )(\dfrac {1 - \tan \dfrac {\phi}{2}}{1 + \tan \dfrac {\phi}{2}})$.
The required ratio $ = \rho = R^2 \phi - \pi r^2 $.
$\phi_{max}$ can be found by applying the usual procedure as mentioned in the comment.