Plane Regular Polygon in a Sphere

This example demonstrates constructing a regular polygon (e.g., equilateral triangle, square) on a plane inside a sphere, and exploring its rotational symmetry and geometric relationship with the spherical surface. Core mathematical concepts include intersection of a plane and a sphere, rotational symmetry of regular polygons, geometric transformations (rotation, reflection) in 3D space, and the property of a polygon inscribed in a sphere. Construction uses GeoGebra commands such as Circle, Point, Segment, Rotate, Angle, Intersect, Sphere, PointIn, and Polygon. The process involves creating a sphere, defining points on a plane within it, rotating points to generate polygon vertices, and using intersection commands to determine contact points with the sphere. This is suitable for high school or undergraduate solid geometry lessons, helping students visualize how planar figures interact with a sphere in 3D and understand the role of rotational symmetry in geometric construction.