Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains

Let $Ω\subset\mathbb{R}^2$ be a bounded planar set that is star-shaped with respect to the origin, and let $A,B,C,D$ be independent random points uniformly distributed on $Ω$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $Θ\in[0,π/2]$ formed b