Exploring Centroids of Sub-triangles Formed by a Point

This GeoGebra applet explores a classic problem in plane geometry concerning the centroids of triangles. Given a triangle ABC and an arbitrary interior point G, the applet constructs three sub-triangles: ACG, ABG, and BCG. It then precisely locates the centroids of these three sub-triangles, labeled G1, G2, and G3 respectively. The resulting centroid triangle G1G2G3 is formed by connecting these three points. Users can dynamically drag point G or any of the vertices A, B, or C of the original triangle. As the configuration changes, the applet allows real-time observation of the relationships between the original triangle ABC and the newly formed centroid triangle G1G2G3. Key properties include the area ratio between the two triangles, vector relationships among the centroid points, and the positional characteristics of triangle G1G2G3 relative to triangle ABC. The model serves as an interactive visual tool for deepening understanding of barycentric coordinates, vector operations, and polygon geometry. It is particularly valuable in the context of analytic geometry and vector algebra instruction, enabling students to conjecture and verify geometric properties through hands-on exploration rather than relying solely on abstract proofs. By manipulating the positions of point G and the triangle's vertices, learners can discover that triangle G1G2G3 is always similar to a triangle formed by connecting specific points related to G, and that certain area ratios remain invariant regardless of the configuration. This dynamic environment supports inquiry-based learning, allowing students to form hypotheses, test them across multiple cases, and develop intuitive geometric reasoning skills that complement formal deductive methods.