An example

Author:Zoltán Kovács

According to Wikipedia, locus is a set of points whose location satisfies or is determined by one or more specified conditions. Being more specific, in GeoGebra locus is the set of output points P' constructed by given steps while the input point P is running on a certain path. In other words, let point P be an element of a path, and let point P' is the output point for the chosen input P after some transformations of P into P'. In general the locus is a curve as the output set of points P', since also the input points P build up a curve. For example, let the input curve be circle c and P is a perimeter point of c. Let the center of the circle be C. Now let us construct point P' such that P' is the midpoint of PC. Clearly, the locus curve here is also a circle described by center C and half of the radius of c. This example can be entered into GeoGebra either by using the graphical user interface with the mouse, or by the keyboard in the Algebra Input (here we put point C into (2,3) and use radius 4):

C=(2,3)
c=Circle[C,4]
P=Point[c]
P'=Midpoint[P,C]
Locus[P',P]

Now by using LocusEquation[P',P] instead of Locus[P',P] we can also check the result algebraically: an implicit curve is displayed with the equation

. It is also possible to see how the equation changes dynamically when point C is dragged. How can this equation be computed mathematically? Let us define coordinates

and

for the points defined above. Now the following equations are valid:

What we need is to convert this equation system to a single equation containing only coordinates of point P'. In algebra this computation is called elimination, i.e. eliminating all variables except

and

In GeoGebra this computation is achieved by the Giac CAS in the background, but it can also be computed directly by using a GeoGebra command (which calls the appropriate Giac statement):

Here the user can add command LocusEquation[loc1] to make GeoGebra compute the equation automatically. By clicking on the marble to the left of the input line GeoGebra will also display the geometric form of the equation, i.e. another circle will be drawn (the same as loc1).

An example

New Resources

Discover Resources

Discover Topics