Does Segment Bisection Imply Angle Bisection? A 3D Vector Exploration

This GeoGebra example investigates the classic geometric question: if a segment is bisected, does it necessarily bisect the corresponding angle? It demonstrates a three-dimensional construction involving a triangle and vectors, where the relationship between segment bisection and angle bisection is visualized and tested. Core mathematical concepts include segment midpoints, angle bisectors, triangle geometry, and vector properties in 3D space. The construction uses GeoGebra commands such as Polygon to create triangle poly1, LineBisector to construct bisectors, Intersect to find intersection points, and Angle to measure angles β, γ, and α, with points A, B, C, D, E defined. This example is suitable for high school or early college students learning geometry, proof, and counterexamples. By dragging points, learners can observe whether segment bisection always leads to angle bisection, enhancing their understanding of geometric conditions and spatial reasoning.