If you understand slopes well, you can answer the question without much algebra. Recall that perpendicular slopes are negative reciprocals. So if you travel, say, 5 units to the right and 2 units up along one line, then to travel the same distance along a perpendicular line, you would need to travel 2 units to the left and 5 units up (we reverse one direction because the perpendicular slope is negative, and we exchange the horizontal and vertical movements because the slopes are reciprocals). Using that here:

The diagonals of a square meet at their midpoints. The midpoint of (6, 2) and (0, 6) is (3, 4) (averaging the x and y coordinates). That must also be the midpoint of the other diagonal.

The diagonals of a square are also perpendicular. We know on the diagonal from (0, 6) to (6, 2), to go from (0, 6) to (3, 4), we go right 3 units and down 2 units. To travel the same distance from (3, 4) along the perpendicular diagonal, we would need to go left 2 units and down 3 units. That will take us to the point (1, 1), so that must be a vertex of the square.