Locus of Hypotenuse Midpoint of Right Triangle Inscribed in Parabola

This GeoGebra applet demonstrates the locus of the midpoint of the hypotenuse of a right triangle inscribed in a parabola. The construction begins with a standard parabola, typically y equals x squared, on which three points are placed. Two of these points serve as the endpoints of the hypotenuse, while the third point forms the right angle vertex on the parabola. As the right-angle vertex moves along the parabola while maintaining the right angle condition, the positions of the other two vertices adjust accordingly to preserve the inscribed right triangle configuration. The applet dynamically tracks the midpoint of the hypotenuse throughout this motion, revealing that the collection of all such midpoints forms another parabola. Students can interact with the applet by dragging the vertices along the curve, toggling the trace feature to observe the full locus being drawn in real time, and comparing the original parabola with the轨迹 parabola generated by the midpoints. The construction leverages GeoGebra geometric tools to enforce the perpendicularity constraint at the right-angle vertex, ensuring mathematical correctness at every position. This applet supports several important mathematical concepts. It illustrates locus problems within conic sections, showing how a specific geometric condition produces a predictable curve. It connects the geometry of right triangles to the algebra of parabolas, demonstrating that the midpoint locus preserves the parabolic form. It also reinforces coordinate geometry by allowing students to verify analytically that the trajectory is indeed a parabola with a related equation. The dynamic visualization bridges the gap between abstract algebraic derivations and concrete geometric intuition, making it particularly valuable for learners studying analytic geometry and conic section properties. Educators can use this applet as an exploratory tool before introducing formal proofs, or as a verification aid after analytical solutions are derived. By observing the pattern emerge visually, students gain confidence in the result and develop deeper insight into why the midpoint of the hypotenuse traces a parabolic path rather than a generic curve.