[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][br]In the case of a constant sum [b]k[/b] of distances to three points [b]A[/b], [b]B[/b] and [b]C[/b], you simply need to input:[br][br] [color=#CC3300]XA + XB + XC = k[/color][br]  [br]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 [b]h[/b] is a positive real parameter that decreases from the value of [i]k[/i] to zero. The boundary points of color will then be precisely the points that satisfy:[br][br] [color=#CC3300]Ellipse(A, B, (k–h)/2) =  Circle(C, h)[/color][br][br]which is equivalent to the sum of the distances from those points to [b]A[/b], [b]B[/b] and [b]C[/b] being exactly the predetermined quantity [b]k [/b](since XA + XB = k – h, XC = h). This way, we can display a [b]3-ellipse[/b] [url=https://en.wikipedia.org/wiki/N-ellipse][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]. In the case of four points the traces of two ellipses overlap, determining a 4-ellipse.[br][list][*][color=#808080]Note: An algebraic approach to this situation, also using GeoGebra, can be seen in this article [[url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]8[/url]] by Zoltán Kovács.[/color][/*][/list]

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]