10.1 The general idea

One of the essential advantages of dynamic geometry software against pen and paper is the possibility to trace the movement of points. Though this idea did not develop through the use of computers - loci can be found in works of Gauß and other mathematicians - they now can be studied by students, teachers and other people who are not as brilliant as Gauß.

10.1.1 The parabola as a locus

The parabola, nowadays mostly known as the graph of f(x) = x², can be defined by giving its focus and directrix: A parabola is the set of all points P_x that have the same distance to a given point P (focus) and a given line l (directrix).

A parabola can be constructed as a locus as follows:

• Given a line l and a point P.
• Construct a Point X on l.
• Draw the normal to l through X.
• Draw the perpendicular bisector of X and P.
• The intersection of the perpendicular bisector with the normal to l has the same distance to P as to l. This follows from the fact that all points on the perpendicular bisector have the same distance to P as to X, so this is true for the intersection with the normal, too. For points on the normal the distance to X is the same as to l, thus the intersection is the point we were looking for.

Now you can construct the parabola with a DGS of your choice.