Hyperbola Reflection Property: Focus and Orthocenter Visualization

This GeoGebra example dynamically demonstrates a key reflection property of a hyperbola: a ray originating from one focus, after reflecting off a point on the hyperbola, appears to emanate from the other focus. Specifically, the tangent at any point on the hyperbola bisects the angle formed by the segments connecting that point to the two foci, embodying the law of reflection. Core mathematical concepts include the definition of a hyperbola (constant difference of distances to two foci), foci, tangent and normal lines, and the reflection property, with the orthocenter playing an auxiliary role in the construction. Construction steps: use Point to create key points A, B, C; use Hyperbola to draw the curve c; use Line to construct tangents or normals g, h; use Intersect to find intersections E, F; and employ orthocenter-related objects i, j, k to verify the reflection relationship. This example is suitable for high school or early college students to intuitively grasp the optical properties of conic sections, and serves as a teaching resource for geometric construction and dynamic exploration.