Section Formula and Barycentric Coordinates in Triangles

This GeoGebra applet demonstrates the section formula and barycentric coordinates for a point P relative to a triangle ABC. Using vector and matrix operations, it calculates the linear combination coefficients of P with respect to the triangle vertices A, B, and C. The interactive quiz and dynamic verification help learners intuitively understand affine combinations, basis transformations, and the geometric meaning of barycentric coordinates in the plane. The applet presents a triangle defined by three freely draggable vertices A, B, and C, along with a target point P whose position can also be manipulated interactively. When the user adjusts any point, the applet instantly recomputes the barycentric coordinates (alpha, beta, gamma) that express P as a weighted affine combination of the triangle vertices, satisfying alpha + beta + gamma = 1. The coefficients are displayed in real time alongside their geometric interpretation, allowing students to observe how the values change as P moves across the plane. Key features include a step-by-step quiz mode that challenges learners to predict the barycentric coordinates for given configurations, followed by dynamic verification that visually confirms correctness through highlighted regions and proportional area indicators. The applet illustrates that when P lies inside the triangle, all three coordinates are positive, and each coordinate corresponds to the normalized area of the sub-triangle formed by P and the opposite edge. When P lies outside, one or more coordinates become negative, reinforcing the extended geometric interpretation beyond the interior case. Matrix-based computation is shown explicitly, linking the vector equation P = alpha*A + beta*B + gamma*C to a solvable linear system, thereby connecting abstract linear algebra to concrete geometric reasoning. This dual perspective helps students bridge the gap between computational techniques and spatial intuition. The interactive design supports exploration of special points such as the centroid, where all three coordinates equal one-third, as well as midpoints and points on the edges of the triangle, each yielding characteristic coordinate patterns. Through guided interaction and immediate visual feedback, the applet serves as an effective learning tool for high school and undergraduate students studying analytic geometry, vector spaces, and affine geometry.