... AXES¹

©1998 by Berkeley Math Circle, Berkeley, CA

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... paths.²

Another way to see why Inversion reverses angles is to view Inversion as the composition of two functions: $f_1(z)=1/z$ for $z\not = 0$, and the reflection along the $x$-axis, $f_2(z)=\overline{z}$: thus, $I(z)=f_2\circ f_1$. Since $f_1$ preserves angles (it is holomorphic), and $f_2$ reverses angles (simple geometric verification), it follows that their composition $I$ will reverse angles.

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