I'm reading a College Geometry: A Unified Development [unfortunately not available through google book preview], and I came across the Theorem, that circle inversions preserve angles between two arbitrary intersecting curves
The proof goes about proving, that the angle $\theta$ between the curves $C_1$ and $C_2$ (depicted in $\color{red}{\text{red}}$ and $\color{blue}{\text{blue}}$ colors respectively in the figure) is the same as the angle $\theta'$ between the images of those curves under the circle inversion.
$t_1$ and $t_2$ (colored lines) are tangents to $C_1$ and $C_2$ curves respectively.
$t_1'$ and $t_2'$ circles are images of $t_1$ and $t_2$ under the inversion (straight lines are mapped to circles through $O$ – the center of circle of inversion).
Dashed lines near the $O$ are tangents to circles, and the solid lines near the second point of circle intersection ($P'$) – are tangents to mapped curves.
The proof goes by claiming that $\varphi$ – the angle between dashed tangent lines to $t_1'$ and $t_2'$, is equal to $\theta$ (one of the properties, that were previously proved, is that tangent line to the circle at $O$ is parallel to its image under the inversion) [so far so good]
… and the angle $\theta'$ between the tangent lines to the $C_1'$ and $C_2'$
mapped curves equals the angle between the circles ($t_1'$ and $t_2'$)
at point of intersection, which in turn equals to $\varphi$ …
So, once I know that black non-dashed lines are indeed tangent to circles I'm done. But why are those lines tangent to circles?
Best Answer
Since the question asks for a clarification of a textbook proof, I'll post copies of the proof and diagram, because the answer will refer to them.
The question states that "the lines near the second point of circle intersection - are tangents to mapped curves." However, the proof in the textbook says that the straight lines at $P'$ are tangents to the circles $t'_1$ and $t'_2$,which are the images of the original curve tangents. The confusion comes, I believe, from the phrase "corresponding tangents $t'_1$ and $t'_2$ at $P'$ to the image curves $C'_1,C'_2$" Here they mean that the circles are tangent to the curves, and they seem to be making an implicit assumption that if curves touch ("are tangent to each other") then the image curves will also touch ("be tangent").
So it remains to show that the tangents to the image circles are the same as the tangents to the image curves at $P'$. This can be done by treating the tangents as the limits of secants going through $P'$, and it is a fairly straightforward exercise to show that the circle secants and curve secants converge to the same straight line.
There is a traditional (and in my opinion canonical) proof involving secants that most other texts use to show the anti-conformal property of inversions. Wolfe's Introduction to Non-Euclidean Geometry, pg. 240 gives one version. In case this link isn't stable, here's a screen cap: