If the answer is the same for any point within the triangle, then I can pick any point I want and solve for that point.

I choose one of the vertices of the triangle.

The distance to 2 of the sides is zero.

The distance to the other side is just the altitude of an equilateral triangle which is sqrt(3)/2.

average(0,0,sqrt(3)/2) = sqrt(3)/6

I haven't proved that the answer is the same for every point in the triangle, but the statement of the problem implies that I can assume it.  So:  not a rigorous solution, but if this were a question on a standardized test, you could save a lot of time, get the right answer, and move on to the next question.