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link to the referenced picture: http://www.flickr.com/photos/90803347@N03/9220374271/

In order to prove the Arbelos Theorem, as in the picture above, one need to prove that the semicircle $C$ invert to line $l$, as well as $D$ invert to $m$, with respect to the circle of inversion that is centered at point $P$ and orthogonal to the circle $K_n$.

How to prove this?

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Inversion takes circles through the center of inversion to lines. Here (the full circles belonging to) $C$ and $D$ have no point in common except the center of inversion, hence their images have no finite point in common either, i.e. the lines they become are parallel. Also, $K_n$ is invariant under the inversion as it is orthogonal to the inversion circle. Since $C$ and $D$ are tangent, their images are tangent to (the image of) $K_n$. And because $C,D$ are orthogonal to the line $PQ$, their images are orthognonal to the image of $PQ$, which is again the line $PQ$ (because it passes through the center of inversion $P$).

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