The ancient Greeks were able to construct a perpendicular bisector for a given line segment using only a straightedge and compass.

Sure, here is a step-by-step guide on how the ancient Greeks could have constructed a perpendicular bisector for a given line segment using only a straightedge and compass:

1. Draw the given line segment using the straightedge. Let's call the endpoints of this line segment A and B.

2. Open your compass to the length of line segment AB. Place the compass point on point A and draw an arc above or below the line segment.

3. Without changing the compass width, place the compass point on point B and draw another arc so that it intersects with the first arc.

4. The point where the arcs intersect is equidistant from points A and B. Draw a line from this point to the midpoint of line segment AB using the straightedge.

5. This new line is the perpendicular bisector of line segment AB. It divides the original line segment into two equal parts and forms a right angle with the original line segment.