A regular triangle ABC is given. Where to put point D in order to the sum of two distances between it and the vertices of the triangle will be equal to the distance between it and the third vertex?[br][br]GeoGebra can answer this question when using the [code]LocusEquation[f+g==h,D][/code] command. In fact, it computes the answer at the same time also for the conditions g+h==f and h+f==g.[br][br]The answer is the union of two circles. The web version of GeoGebra (as of July 2016) cannot compute the answer because of memory limitations, but the desktop version can:

The polynomial of this algebraic curve is a product of two circles, but the y-coordinate of their centers is [math]\frac{\sqrt{3}}{6}[/math]. That prevents having a factorization with integer coefficients only.[br]"Factorization" of algebraic curves can be crucial when plotting implicit curves. Plotting the factors separately increases the quality of the plot. In this case, however, factorizaton is not really required: both circles can be seen perfectly.[br]Absolute factorization (that is a factorization over an algebraic closure of its coefficient field) is not yet implemented in GeoGebra. It's available in some other computer systems like [url=https://www.maplesoft.com/support/help/maple/view.aspx?path=AFactor]Maple[/url] or [url=https://www.singular.uni-kl.de/Manual/4-0-3/sing_1136.htm]Singular[/url].

[url=https://www.geogebra.org/m/UTB5tBpp]Holfeld's 24th problem[/url] in his [url=http://home.pf.jcu.cz/~hasek/Holfeld/]Exercitationes Geometricae[/url] (1773) also produces a locus equation which can be factorized, but not always over the integers. Try to drag point B to various points, e.g. to (0,4).