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When defining an Apollonius circle, in $\mathbb{R}^2$, 5 parameters are required: $(x_A, y_A), (x_B, y_B)$ and the ratio $\lambda$. But for circles’ ordinary definition, only 3 parameters: center location $(x_c, y_c)$ and radius $r$ are needed. Different numbers of parameters return the same result, why?

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    $\begingroup$ Because you can obtain the SAME Apollonius circle with infinitely many different parameters. $\endgroup$ – Aretino Sep 22 '17 at 17:31
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An Apollonius circle is defined as the locus of the points $P$ such that the ratio $PA/PB$ of the distances from two fixed points $A$ and $B$ is a given positive number $\lambda$.

Given a circle, however, there are infinitely many triads $(A, B, \lambda)$ having that circle as their Apollonius circle.

Consider, for example, the circle in the cartesian plane with equation $x^2+y^2=1$: it is the Apollonius circle of all $(A,B,\lambda)$ with $A$ and $B$ given by: $A=(\lambda,0)$, $B=(1/\lambda,0)$. And if you rotate each couple $(A,B)$ around the circle center, the rotated points $(A',B')$ define the same Apollonius circle with the same ratio $\lambda$.

In summary, that circle is the Apollonius circle of all the triads: $$ A=(\lambda\cos\theta,\lambda\sin\theta),\quad B=\left({\cos\theta\over\lambda},{\sin\theta\over \lambda}\right),\quad \lambda. $$ These depend on two free parameters $\theta$ and $\lambda$: that explains the difference between five parameters of Apollonius' definition and three parameters of the ordinary definition.

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  • $\begingroup$ [+1] nice answer ! Another kind of answer would take advantage of the fact that (A,B;C,D) form an harmonic division (where C,D are the points of intersection of the circle with line AB) $\endgroup$ – Jean Marie Sep 22 '17 at 22:40

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