I started the construction by drawing segment AB, centering two circles around the endpoints and added segment CD to the intersection points of the circles. I then drew segment GH at the bottom of the circles and connected them to make a segment without intersecting the circles. I then added two circles to the endpoints and added segments NR,HB,AQ, EA,BF, HI, GL and PQ. BQ is parallel to HA because the perpendicular segments NR and PQ overlap them and are parallel to the perpendicular bisector because they are perpendicular to the same line. BH and QA are parallel because they are both perpendicular to MD. HA=BQ because they are radii of the same circles. BH=QA because they go from the center of the circle to the bottom edge of the same circles.