Napoleon's Triangles (Triángulos de Napoleón) Dynamic Geometry

①This example demonstrates Napoleon's theorem: given an arbitrary triangle, construct equilateral triangles outward (or inward) on each side; the centroids of these three equilateral triangles form a new triangle that is always equilateral. ②Core mathematical concepts include equilateral triangles, centroid (barycenter), vectors, midpoints, angles, and classical triangle geometry theorems. ③Construction highlights: using GeoGebra commands such as Circle, Point, If, Polygon, Centroid, Vector, Midpoint, and Angle, the applet builds the original triangle (polygon1), three equilateral triangles (polygon2, polygon3, etc.), centroid points (G, H, etc.), and the Napoleon triangle connecting these centroids; dynamic sliders or draggable points allow the user to change the original triangle and observe that the Napoleon triangle remains equilateral in real time. ④Suitable for high school and above, or geometry enthusiasts, for exploring triangle properties, dynamically verifying theorems, and as a classroom demonstration or math experiment.