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You are given two points and a circle. Construct a circle passing through the given two points and tangent to the given circle. You are allowed to use a straightedge and a compass.

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marked as duplicate by Blue geometry Apr 25 '16 at 10:16

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Here's a "birds.view" description of a workable method: Let $A,B$ be the points and $c$ the circle. Perform an inversion that takes $A$ to infinity (and $B$ to $B'$ and $c$ to circle $c'$). Now construct the two tangents $\ell_1,\ell_2$ to $c'$ through $B'$. Invert back, which takes $\ell_{1,2}$ to circles that pass through $A$ and $B$ and touch $c$. - Some degenerate cases are possible, e.g., one of the final circle may in fact be a line ...

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