Geometric Construction of a Square

This applet demonstrates the classical geometric construction of a square using GeoGebra's dynamic geometry tools. The process begins by constructing a given line segment, which serves as one side of the square. From each endpoint of this segment, the user constructs a perpendicular line using the standard compass-and-straightedge technique. Next, circles are drawn centered at each endpoint with a radius equal to the length of the original segment. These circles intersect the corresponding perpendicular lines at points that are exactly one side length away from the endpoints. The intersection points define two additional vertices of the square. By connecting all four vertices in sequence, a complete square polygon is formed. The applet allows users to dynamically manipulate the initial segment—dragging its endpoints to change its position, length, or orientation—and observe how the entire construction adapts in real time while preserving all geometric relationships. This interactive feature reinforces the understanding that the properties of a square, including equal side lengths and right angles, are invariant under rigid transformations. The step-by-step construction visually highlights fundamental geometric principles such as perpendicularity, congruence of segments, and the role of circle intersections in locating precise points. It is particularly well suited for introductory平面几何 courses, providing students with a clear, visual proof that a square can be constructed from a single segment using only a compass and an unmarked straightedge. The applet supports both guided exploration and independent experimentation, making it a valuable teaching tool for helping learners internalize the logic and rigor of classical Euclidean constructions.