Focus of a Parabola: Geometric Construction and Properties

This GeoGebra example dynamically demonstrates the geometric meaning of the focus of a parabola and its relationship with tangents and normals. The core mathematical concept is the conic section definition of a parabola—the locus of points equidistant from a fixed point (focus) and a fixed line (directrix)—along with the reflective property: rays from the focus reflect off the parabola parallel to its axis. The construction uses commands such as Segment to create segments, Tangent to draw the tangent line, OrthogonalLine to construct the normal, Mirror to reflect points across the normal, Line to connect points, and Intersect to find intersection points. Key objects include the parabola a, focus f, points A and B, tangent g, normal h, mirror line i, reflected point A', line k, and intersection point C. This example is suitable for high school analytic geometry and conic sections lessons, as well as introductory college geometry, helping students visualize the optical property of parabolas and appreciate the power of dynamic geometry software in exploring mathematical relationships.