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I'm reading Conway complex analysis book and on page 48 he proves the following proposition:

Proposition. If $z_2$, $z_3$, $z_4$ are distinct points in $\mathbb{C}_\infty$ and $\omega_2$, $\omega_3$, $\omega_4$ are also distinct points in $\mathbb{C}_\infty$, then there is one and only one Möbius transformation $S$ such that $S z_2 = \omega_2$, $S z_3 = \omega_3$, $S z_4 = \omega_4$.

Afterwards he made the following claim:

It is well known from high school geometry that three points in the plane determine a circle. (Recall that a circle in $\mathbb C_{\infty}$ passing through $\infty$ corresponds to a straight line in $\mathbb C$. Hence there is no need to inject in the previous statement the word "non-collinear.") A straight line in the plane will be called a circle. The next result explains when four points lie on a circle.

I did understand everything he says except: "Hence there is no need to inject in the previous statement the word 'non-collinear'."

Could someone clarify this statement?

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