Invariance of the Sum of Areas: A Dynamic Geometry Demonstration

① This GeoGebra example demonstrates a geometric phenomenon where the total area of multiple polygons remains invariant under certain transformations, such as rotation or subdivision. ② Core mathematical concepts include area conservation, polygon decomposition and recombination, rotational symmetry, and connections to upper/lower Riemann sums, providing an intuitive model for area accumulation in integral calculus. ③ Construction highlights: using GeoGebra commands Polygon, Midpoint, Segment, and Intersect to build the base figure; objects include points A, B, C, D, polygon poly1, segments a, b, c, d, midpoints E, F, G, H, intersection O, and auxiliary polygons S_1 through S_4, along with function-related objects, enabling dynamic rotation or subdivision to illustrate the invariant sum. ④ Suitable for upper secondary to early university students, this example serves as a visual aid in geometry teaching, an introduction to pre-calculus concepts like Riemann sums, and a dynamic exploration tool. By dragging or animating, learners can observe the constancy of the area sum, deepening their understanding of area conservation and transformation invariance.