Hyperbola Lab: Construction via Pythagorean Theorem and Vector Rotation

① This example demonstrates how to construct a hyperbola using circles and intersection points, while revealing its geometric essence through vector rotation and the Pythagorean theorem. ② Core mathematical concepts include the definition of a hyperbola (constant difference of distances to two foci), the Pythagorean theorem in distance calculations, 2D vector rotation, and fractal geometry iteration. ③ Construction highlights: using GeoGebra's Circle command to draw auxiliary circles, the Intersect command to obtain points A1, A2, K, A, B, C, D, E, F, constructing lines c, d, e, f from these points, and applying vector rotation to generate dynamic loci that form the hyperbola. ④ Suitable for high school or early college students to explore conic section geometry, vector transformations, and fractal patterns, and as a resource for math lab activities or advanced GeoGebra practice.