Parabola: Visualizing Geometric Definition and Distance Comparison

This GeoGebra example demonstrates the geometric definition of a parabola by dynamically comparing the distance from a moving point $M$ to the focus $F$ and to the directrix $d$, visually verifying their equality. The core mathematical concepts involved include the geometric definition of a parabola (focus, directrix, eccentricity), the distance from a point to a line, and the locus concept in coordinate geometry. To construct this, the example utilizes GeoGebra commands such as `Point` to define the focus $F$, the moving point $M$, and other key elements; `OrthogonalLine` to construct perpendiculars; `Intersect` to find the projection $H$ on the directrix; and `Segment` to draw the comparative distance segments. Key objects include the parabola $p$, axis $a$, focus $F$, and directrix $d$. This resource is ideal for high school algebra/precalculus or introductory college-level coordinate geometry courses to visualize conic section definitions.