Triangle Centroid Property: Midpoints, Vectors, and Transformations

This GeoGebra example demonstrates the centroid property of a triangle by showing how the centroid relates to vertex connections, midpoints, and vector directions. It centers on key concepts: the centroid of a triangle (Centroid), representing geometric directions with vectors (Vector), constructing and connecting midpoints and segments (Midpoint, Segment), and validating relationships using geometric transformations such as reflection, rotation, and dilation (Mirror, Rotate, Dilate). Construction uses Polygon to create the triangle, computes the centroid point G with Centroid, and then draws essential lines with Segment while using Vector to indicate direction/translation relationships involving G and related points (e.g., points tied to parameters like t1). The commands Mirror, Dilate, and Rotate generate transformed configurations to visually confirm the centroid property and its geometric consequences. Point objects and labels (including A, B, C, G, P and several auxiliary points) help link the theory to the constructed diagram. Suitable for secondary school through early university, especially for geometry construction, vector-based reasoning, and transformation-assisted proofs.