A theorem states, that the three Apollonian circles, associated with the given triangle $ABC$ with sidelengths $a \neq b \neq c$ intersect in two points. The proof proceeds by showing that if the two circles passing through $A$ and $B$ intersect at some point $P$, then the third one also passes through $P$. But how does one prove that the former two circles intersect in the first place?
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