Ellipse Optical Property: Constructing Reflection via Tangents and Foci

This example demonstrates the optical property of an ellipse: a ray (modeled by a line segment) starting from one focus reflects off the ellipse so that it passes through the other focus. It visually verifies the reflection behavior using geometric constructions rather than relying on purely algebraic computation. Key concepts include the ellipse as a conic section defined by its foci, tangent geometry, and the reflection law (the angle of incidence equals the angle of reflection), implemented through angle measurement and construction by slopes/angle relations. Construction highlights in GeoGebra: Points are created for the foci and a chosen reflection point; Focus specifies the two ellipse foci; Mirror and Tangent establish the reflection relative to the tangent line; PolyLine/Segment draw the incoming and outgoing rays; OrthogonalLine provides helper perpendiculars; Intersect locates critical intersection points; Angle measures the relevant angles to confirm equality; Polygon and related conic/curve objects organize the scene. The construction uses objects such as P, F_1, F_2 and auxiliary points/parameters (e.g., α, β, a, b, c and intersection-defined points) to connect the ellipse’s geometry with its optical interpretation. Suitable for advanced high school through undergraduate students studying analytic geometry, conic sections, and geometric optics-style interpretations.