mg.metric geometry - How to compute the average distance till intersection within a triangle in R^2?

We can start by solving the problem for an arbitrary triangle and a fixed direction, say the upward direction, as we can always rotate the whole figure into that position.

In the generic case, no edge of the triangle is horizontal. Thus the triangle has a "top" vertex at (0, 0) -- this is the vertex with the largest y-coordinate. Without loss of generality, rescale so that the edge opposite the vertex at (0, 0) passes through (-1, 0). We'll say that the "height" of this triangle is 1, where by height we mean the length of the vertical dropped from the top vertex.

Then the set of points of vertical distance at least r from the top of the original triangle form a triangle of height 1-r, similar to the original triangle. Thus

$$ Prob(hbox{vertical distance from top } > r) = (1-r)^2 $$

and this gives the distribution. In particular the expected value of the vertical distance from a random point to the top of the triangle (after the rescaling given above) is 1/3.

If you want to pick a random direction, though, then this gets a lot harder because the rescaling step will act differently for different directions.