[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]When the [b]sum [/b]of the distances from the points of the sought-after locus to points [b]A[/b] and [b]B[/b] is constant, we obtain an ellipse.[br]  [br]When the [b]difference [/b]of the distances from the points of the sought-after locus to points [b]A[/b] and [b]B[/b] is constant, we obtain a branch of a hyperbola. (In the case where the constant is 0, we obtain the perpendicular bisector.)[br][br]When the [b]product [/b]of the distances from the points of the sought-after locus to points [b]A[/b] and [b]B[/b] is constant, we obtain a [i]Cassini oval[/i] [url=https://en.wikipedia.org/wiki/Cassini_oval][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]. If the constant coincides with the square of half the distance AB, we obtain a [i]Bernoulli lemniscate[/i] [url=https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]. [br] [br]Surprisingly [url=https://www.gaussianos.com/la-sorprendente-aparicion-de-una-curva-muy-conocida/][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], , when the [b]quotient [/b]of the distances from the points of the sought-after locus to points [b]A[/b] and [b]B[/b] is constant, we obtain a circle. (In the case where the constant is 1, we obtain the perpendicular bisector.)[br][list][*]Note: For a better view of the construction, it is recommended to download the ggb file [url=https://www.geogebra.org/material/download/format/file/id/xvykvefe]here[/url].[/*][/list]

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