Pythagorean Theorem Generalization with Semicircles (Example 5)

This example demonstrates a generalization of the Pythagorean theorem using semicircles, where the areas of semicircles on the sides of a right triangle satisfy the classic theorem. ① What the example demonstrates: The applet visualizes a semicircle-based extension of the Pythagorean theorem, showing how the areas of three semicircles relate in a right‑angled triangle. ② Core mathematical concepts: The key concepts include the Pythagorean theorem, semicircle area relationships in plane geometry, coordinate‑geometry constructions of points and segments, and limit ideas (the relationship holds as radii become arbitrarily large). ③ Construction highlights: Built with GeoGebra commands such as Midpoint, Polygon, Segment, Intersect, Rotate, Line, OrthogonalLine, Angle, and CircleSector. Points A, B, C, D are defined, squares and semicircles with radii a, b, c are created, and animation buttons and text objects dynamically display the area equality. ④ Suitable grade level or use: Ideal for high‑school mathematics (geometry, analytical geometry) and university introductory courses. It can be used to teach geometric proofs of the Pythagorean theorem, explore area relationships, and provide dynamic visual demonstrations that enhance spatial reasoning and analytical skills.