Let $ABC$ be a triangle with incenter $I$, and let the tangency points of its incircle with its sides $AB$, $BC$, $CA$ be $C'$, $A'$ and $B'$ respectively. Prove that the circumcenters of $AIA'$, $BIB'$, and $CIC'$ are collinear.
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Tags: geometry, incenter, circumcircle
Let $ABC$ be a triangle with incenter $I$, and let the tangency points of its incircle with its sides $AB$, $BC$, $CA$ be $C'$, $A'$ and $B'$ respectively. Prove that the circumcenters of $AIA'$, $BIB'$, and $CIC'$ are collinear.