As title suggests, in the figure given below with a square, a semicircle and some unknown angles $\alpha$ and $\theta$, the goal is to find the numerical value of the expression: $$\frac{\alpha}{\theta}$$
I'm not entirely sure how to go about this problem, its a fairly unique type of problem that I've never encountered before. Nevertheless, I'll also post my own approach below, please share your own as well!
This is my approach. I'll add an explanation below as well:
Here's my explanation:
1.) Connect point $C$ and $A$ via the diagonal $AC$. Next, also, connect the tangency point $T$ with the point of intersection of both diagonals (let's call it point $X$). Lastly, connect the tangency point $T$ with $A$. Using the properties of cyclic quadrilaterals and some circle theorems, we can deduce that $\angle BTA=\angle BXA=90$. And also that $\angle TBX=\angle TAX=\alpha$.
2.) Notice that since $DT$ is tangent to the semicircle, it is equal to segments $DA$ and $DC$ (all the sides of the square). This means that via the inscribed angle theorem, $\angle TDC$ is twice the measure of $\angle TAX$. Thus $\angle TDC=2\alpha$. This also implies that $\angle TDA=90-2\alpha$. Once again, via the inscribed angle theorem, this means that $\angle TCA$ is half of $\angle TDA$, thus $\angle TCA=45-\alpha$. However, since $AC$ is a diagonal and $\angle TCB=\theta$, we also know that $\angle TCA=45-\theta$. This implies that:
$$45-\alpha=45-\theta$$
and thus:
$$\frac{\alpha}{\theta}=1$$