Parabolas: Focus, Directrix, and Geometric Properties

①This GeoGebra example dynamically demonstrates the standard equation, focus, directrix, vertex, and focal diameter of a parabola, visually illustrating the definition and properties of a parabola. ②Core mathematical concepts include the focus-directrix relationship, vertex coordinates, focal parameter (p), focal diameter (the chord through the focus perpendicular to the axis), and intersections of auxiliary triangles, quadrilaterals, and kites with the parabola. ③Construction details: The example uses the Segment command to construct segments, with objects such as h, k, p, c, d, Vertex_2, a, b, Focus_1, Focus_2, text1 through text6, B, C, FocalDiameter, Vertex_1, f, D, E, FocalWdth, and g. Sliders or parameters control the parabola's shape, and text objects label key values to show dynamic changes of focus, directrix, vertex, and focal diameter. ④This is suitable for high school analytic geometry, introductory college geometry, or self-learners, serving as an interactive tool for understanding the definition, parameters, and geometric construction of parabolas, and for exploring intersections with triangles and quadrilaterals in classroom demonstrations or student investigations.