I want to construct a quadrilateral inscribed in a circle of known radius $R$ (cyclic) and at the same time I want this quadrilateral to be tangential to another circle of a known radius $r$. In addition to $R$ and $r$, I know one of the side lengths.
I have attempted solving this problem numerically by creating the circumcircle of the quadrilateral and placing the given side as a chord. Then I placed the inner circle (to which the quadrilateral is tangential) tangent to the chord, and using the relative position of the inner circle along the given side, and Newton's iteration, I was able to find the quadrilateral.
Question: Is there a method that does not require trial-and-error to construct such a quadrilateral?
The following image shows an example
where $R = 12 $ , $ r= 7 $ and $ s = 12 \sqrt{2} $
Let $c_R,c_r$ be the circles with radii $R,r$ respectively, $O_R,O_r$ the respective centers, and $d$ be the distance between $C_R$ and $C_r$.
Then, by a formula dating back to Fuss(1798), $$ \dfrac{1}{(R-d)^2}+\dfrac{1}{(R+d)^2}=\dfrac{1}{r^2}. $$
Given $R,r$ we can solve for $d$: $$ d=\sqrt{r^2 + R^2- \sqrt{r^2(r^2 + 4R^2)}}. $$
So your construction can start, as you've done, with circumcircle $c_R$ and chord $s$ of the given length. Construct line $\ell$ parallel to $s$ at a distance $r$ from $s$. Construct circle $\delta$ with radius $d$ and center $C_R$. Let $p$ be one of the points of $\ell\cap\delta$. Then $c_r$ is the circle with center $O_r=p$ and radius $r$. You can now construct the other three sides of the quadrilateral.