Rotating Regular Polygons: Matrix Transformations and Geometric Construction

① This example demonstrates how to generate a series of regular polygons through rotation and observe the changes in position and shape after rotation. ② Core mathematical concepts include properties of regular polygons, rotation transformations (matrix representation), composition of geometric transformations, and the relationships of incircles, orthocenters, and other geometric elements under rotation. ③ Construction highlights: Using GeoGebra commands Circle, Rotate, and Polygon, objects A, B, C, D are created; sliders n (number of sides) and s (rotation angle) control the polygon poly1 and its rotated copy poly1'; points D' and D'_1 mark rotated vertices, while auxiliary circles and orthocenter-related elements are constructed. ④ Suitable for high school or early college students learning geometric transformations and matrix applications, also useful for dynamic demonstrations of rotational symmetry and invariant properties of regular polygons.