A parabola is the set of all points equidistant from a fixed line, called the directrix, and a fixed point, called the focus. Using Geometer's Sketchpad, we can construct a parabola given a fixed point and line by tracing a point that is equidistant from both:
First we begin with the directrix, focus (point F) and some arbitrary point P on the directrix:
A point that is equidistant from the focus and P lies on the perpendicular bisector of segment FP:
Next, we can construct a perpendicular line to the directrix through P that will intersect the perpendicular bisector of segment FP:
If we construct the segment with the focus and the intersection of the two perpendiculars, an isosceles triangle is formed; therefore this point (point L) is equidistant from the focus F and the directrix.
Tracing point L as point P moves along the directrix creates the set of points that represents a parabola.
Notice that the perpendicular bisector of segment FP is tangent to the curve. If we trace this line, this will also create our parabola:
In Geometer's Sketchpad, the locus command can also be used to generate the parabola. If point P and point L are selected, the set of points is created by choosing "Locus" from the Construct menu:
