Line a crosses the line segment BC in the middle. Prove that points B and C are at the same distance from line a.

For the proof, we just need to recall the definition of the midpoint of a segment. The middle of any segment is a point equidistant from each of its ends, that is, dividing the existing segment into two equal parts. If line a intersects the segment BC in the middle, then points B and C will be equidistant from the intersection point by definition. If the straight line a intersects the segment BC at a right angle, then the points B and C will also be equidistant from any other point on the straight line a.