Sum to Three or Four Points
This activity belongs to the GeoGebra book GeoGebra Principia.
In the case of a constant sum k of distances to three points A, B and C, you simply need to input:
XA + XB + XC = k
To perform the offset, what we do is overlay the trace of ellipses Ellipse(A, B, (k–h)/2) with circles Circle(C, h), where h is a positive real parameter that decreases from the value of k to zero. The boundary points of color will then be precisely the points that satisfy:
Ellipse(A, B, (k–h)/2) = Circle(C, h)
which is equivalent to the sum of the distances from those points to A, B and C being exactly the predetermined quantity k (since XA + XB = k – h, XC = h). This way, we can display a 3-ellipse 
. In the case of four points the traces of two ellipses overlap, determining a 4-ellipse.
. In the case of four points the traces of two ellipses overlap, determining a 4-ellipse.
- Note: An algebraic approach to this situation, also using GeoGebra, can be seen in this article [8] by Zoltán Kovács.
Author of the construction of GeoGebra: Rafael Losada.