Lots of simple questions because I am a noob.

You are given 3 points in 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 distr. for the convex polygon?

Finally, how about a non-convex polygon?