Hyperbola Reflective Property – Tangent and Foci Angle Equality

This example demonstrates, without verbal explanation, that a line tangent to a hyperbola makes equal angles with the segments joining the point of tangency to both foci—a fundamental reflective property of hyperbolas. The core mathematics involves hyperbolas, tangents, foci, angle equality, distance calculations, and geometric constructions. Using GeoGebra commands such as Point, Ray, Midpoint, Intersect, Hyperbola, Distance, Circle, Segment, Tangent, CircleSector, Angle, Dilate, and Rotate, the applet builds the hyperbola, its foci, a variable tangent, and measures the angles to verify the property. It is intended for high‑school or early‑university students studying analytic geometry and conic sections, providing an interactive visual proof of the reflective property.