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 a line segment: Using the straightedge, draw a line segment AB.

2. Draw circles with the same radius: Open your compass to the length of AB. Then, draw a circle with center A and another circle with center B. Make sure the circles intersect each other.

3. Identify intersection points: The two circles will intersect at two points. Let's call these points C and D.

4. Draw the perpendicular bisector: Using your straightedge, draw a line through points C and D. This line is the perpendicular bisector of AB. It cuts the original line segment into two equal parts at a 90-degree angle.

This method is known as the "perpendicular bisector construction" and was used by the ancient Greeks for various geometric constructions.