I am reading Howie's Complex Analysis. There I see this remark:
The observation that $c$ and $d$ are inverse points is the key to showing that every circle can be represented as $\{z:|z-c|=k|z-d|\}$. Suppose that $\Sigma$ is a circle with centre $a$ and radius $R$. Let $c = a + t$, where $0 < t < |a|$, and let $d=a+\frac{R^2}t$. Then $c$ and $d$ are inverse points with respect to $\Sigma$. For every point $z=a+Re^{i\theta}$ on $\Sigma$, $$\left|\frac{z-c}{z-d}\right|=\frac{t}{R}$$ and so $|z - c| = (t/R)|z - d|$. The answer is not unique.
I don't understand why $0 < t < |a|$ is assumed? I think it would make more sense to take $0 < t < R$.