Sam's Tetrahedron: Minimum Volume Tetrahedron from Tangent Plane on an Ellipsoid

This GeoGebra example demonstrates the problem posed by Sam Shah: given an ellipsoid x²/A² + y²/B² + z²/C² = 1, choose a point D on the surface, construct the tangent plane at D, and form the tetrahedron bounded by this tangent plane and the three coordinate planes. The goal is to explore the minimum volume of such tetrahedra as D varies. Core mathematical concepts include tangent planes to implicit surfaces, partial derivatives, volume of tetrahedra in 3D space, and optimization. The construction uses GeoGebra commands: PointIn to place D on the ellipsoid, Derivative to compute partial derivatives, OrthogonalPlane to create the tangent plane, IntersectPath and Intersect to find intersections with coordinate axes, Pyramid to build the tetrahedron, and Textfield/InputBox for interactive adjustment of parameters A, B, C. This is suitable for university-level calculus, multivariable differentiation, or spatial geometry courses, and serves as a vivid visualization for optimization problems and tangent plane applications.