Locus of the Centroid of a Triangle Related to a Parabola

This applet demonstrates the locus problem involving the centroid of a triangle associated with a parabola. The construction begins by placing two fixed vertices of a triangle and selecting a third vertex that moves freely along a given parabola. Using GeoGebra's geometric tools, the midpoints of the triangle's sides are constructed, and their intersections are used to locate the centroid — the point where all three medians of the triangle meet. As the dynamic point is dragged along the parabola, the applet continuously updates the triangle and recalculates the position of its centroid in real time. Simultaneously, the trajectory of the centroid is traced, revealing the shape of its locus. The result is a smooth curve that can be compared against the original parabola, allowing students to observe how the centroid's path relates to the geometry of the generating conic section. This interactive visualization provides several key learning benefits. First, it reinforces the definition and properties of the centroid, showing concretely that the centroid always lies two-thirds of the way along each median from a vertex. Second, it deepens understanding of locus problems by letting students see how a dependent point traces a new curve when its defining parameters vary continuously. Third, it connects the algebraic equation of a parabola to its geometric behavior, illustrating that the locus of the centroid in this configuration is itself a parabola — a transformed version of the original. Students can explore how changing the positions of the fixed vertices or the shape of the parabola alters the traced locus. This hands-on experimentation encourages hypothesis formation and verification, promoting an active rather than passive approach to learning conic section theory. The applet is particularly useful for courses covering analytic geometry, coordinate proofs, or the geometric properties of conics, offering an intuitive bridge between abstract equations and concrete visual representations.