A Sphere and the Pythagorean Theorem

This GeoGebra example demonstrates the Pythagorean theorem in three-dimensional space using a sphere and associated segments and polygons. It visually explores how the theorem extends from planar right triangles to spatial configurations involving a sphere's radius and distances between points. Core mathematical concepts include the sphere equation, spatial distance, right-triangle relationships, and intersections of geometric objects. The construction employs GeoGebra commands such as Intersect to find intersection points, Point to define vertices, Sphere to create the sphere, Segment to draw edges, and Polygon to form faces, with auxiliary objects (a, b, c, t1, t2, etc.) linking the geometry. Suitable for high school and college students studying solid geometry or vector algebra, this example serves as an interactive tool for teaching the spatial generalization of the Pythagorean theorem and enhancing spatial reasoning.