Locus of points $P$ on the plane such that $\overline{AP}=\lambda \cdot \overline{BP}$

Question

Given two points on the plane $A$ and $B$ and given $\lambda \in (0,+\infty)$ consider the the locus of all the points $P$ such that $\overline{AP}=\lambda \cdot \overline{BP}$. If you study it with analytic geometry you will end up with the equation of a circle (degenerating to a stright line when $\lambda=1$).

• Question: is there a way to see that this locus is a circle only by geometrical arguments?

Answer 1

This is the well known Apollonius circle whose locus is obtained by squaring and simplification of the given condition.

The result sought uses methods are of geometry and analytical geometry. Also by considerations of Cross Ratio = -1 by harmonic division the locus can be defined identifying nearest and farthest points of this locus.