You are given 3 points in $\mathbb{R}^2$; $A$, $B$, $C$ forming a triangle with area > 0. You pick an arbitrary point inside $ABC$ and an arbitrary direction. After some distance $d$, you will intersect some side of a triangle. The task is to compute the expected $d$ for a given triangle. Also, it would be nice to know the whole distribution. What kind of distribution (e.g. uniform, normal) would it be?

Next, what if I have a complex polygon? Can I combine my knowledge from individual triangles somehow to compute mean and distribution for the convex polygon?

Finally, how about a non-convex polygon?