[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 conclusion, the field to explore can expand indefinitely. As final examples with distances, let's observe some results involving powers.[br][br]It is easy to demonstrate that the representation of XA2 + XB2 = k, with k constant, is a circle centered at the midpoint of A and B.[br][list][*][color=#808080]Note: The radius of that circle is sqrt(k/2 -((x(A)+x(B))²+(y(A)+y(B))²)/4).[/color][br][/*][/list]From this, we deduce that the locus where the sum of the squares of distances to several points is constant is a circle centered at the midpoint of those points.[br][br]Furthermore, taking [b]D = XA2[/b], we can observe that the real-plane representation of any polynomial [b]p(D)[/b] is exclusively composed of one or more circles.[br][list][*][color=#808080]Note: This follows from the Fundamental Theorem of Algebra, since p(D) can be decomposed by factors (D − c), where c is a complex number. If c is non-negative real, then D − c = 0 corresponds to a circle with radius the square root of c. Otherwise, nothing is displayed.[/color][br][/*][/list]Here, we also see that we can represent multiple curves of the same family, such as XA[sup]n[/sup] = XB and observe their behavior [b]simultaneously[/b].

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