If "m" is the slope of the middle line (I am assuming you mean the slope in the "Cartesian geometry" sense and not how the road slopes going up a hill!) then the slope of the line perpendicular to it is -1/m. If the coordinates of the point you are given are , then the line perpendicular to the "road" is given by . The two desired points lie at width/2 from and so lie on a circle with center at and radius width/2. Such a circle has equation .
Replace y in that equation with from the first equation and you have
Take + for one point and - for the other. Of course, y for each point is given by .