Geometric Construction of a Hyperbola from Foci and Distance Difference

This example demonstrates how to construct a hyperbola in a dynamic geometry setting by starting from its foci and enforcing the constant difference of distances. It highlights the geometric character of conic sections: the hyperbola is the locus of points whose distances to two fixed foci differ by a prescribed constant (often denoted via parameters such as 2a). The construction emphasizes core ideas including foci-based definitions, symmetry with respect to the center/axes, and the role of auxiliary lines and circles to locate points on the curve. In GeoGebra, the workflow uses Point to create key points, Mirror to reflect points and exploit symmetry, Hyperbola to generate the conic model, Intersect to compute intersections that determine critical locations, Segment to represent measurable constructions (distance-related segments), and Circle and Line as guiding auxiliary geometry. The involved objects include F_1, F_2 (foci), O (center), and additional constructed points/parameters (e.g., T, r_1, r_2, and intersection-defined points like B, C, D, E) to complete the construction logic. Suitable for high school through early university courses on analytic geometry and conic sections, as well as for student exploration of geometric constructions.