The Parabola as the Set of Points Equidistant from a Line to a Fixed Point
A parabola can be defined as the set of points equidistant from a given line, called the directrix, and a fixed point, called the focus. Let’s begin with the directrix, the focus, F, and variable point, P.
A point that is equidistant from the focus and P is on the perpendicular bisector of FP. Construct the perpendicular bisector of FP.
Constructing the perpendicular to the directrix through P will create an intersection, I, with the perpendicular bisector FP. Construct the segment IF. An isosceles triangle is formed. Thus, point I is equidistant from the focus and the directrix.
Tracing point I as point P moves along the directrix creates the set of point equidistant from the directrix to F. Click HERE to view this animation.
As you can see, the set of points equidistant from a line to a fixed point is a parabola.
In the picture above you can see that the perpendicular bisector of FP is tangent to the parabola at point I. Lets trace the perpendicular bisector as P moves along the directrix. Click HERE to view this animation.
Tracing the perpendicular bisector of FP generates the envelope of the parabola.
You can also generate the parabola using the locus command in GSP. Highlight P and I. Under the Construct menu click Locus.
So, the parabola represents the set of points equidistant from a line to a fixed point. This can be represented using GSP as done above.