In triangle ABC
let side BC be of length a
let side AC be of length b
let side AB be of length c
let P be a randomly chosen point inside ABC with uniform distribution

What is the probability in terms of a,b,c that P will be:

1) perpendicularly closer to side AB than it is to either of the other two sides?
2) perpendicularly further from AB than the combined distances to the other two sides?

Equal?!