Find the Midpoint: Circle and Segment Construction

This GeoGebra example demonstrates how to construct and identify the midpoint of a line segment using the intersection of circles and the segment itself. The core mathematical concept is the definition of a midpoint and the property of a circle as the set of points equidistant from a center. The construction uses the classic compass-and-straightedge method: draw two equal-radius circles centered at the segment's endpoints; the line connecting their intersection points crosses the segment at its midpoint. Key construction steps involve the Circle command to create the two circles, the Point command to mark intersection points, and the Segment command to draw the original segment and highlight the midpoint. This example is suitable for middle school geometry lessons on basic constructions and the midpoint theorem, as well as for high school coordinate geometry to visually verify the midpoint formula. By dragging the endpoints, students can observe how the midpoint moves dynamically, reinforcing the concept of geometric invariance.