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Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$. Let $D,E,F$ be the touchpoints of the incircle with $BC,CA,AB$ respectively. Prove that, $OI$ is the Euler line of $\triangle DEF$.

Consider an inversion with respect to the incircle of $\triangle ABC$. This transformation sends the circumcircle of $\triangle ABC$ to the nine-point circle of $\triangle DEF$.It is well-known that $I,O$ and $O'$ are collinear, where $O'$ is the nine-point centre of $\triangle DEF$. But $I$ is the circumcentre and $O'$ is the nine-point centre of $\triangle DEF$,which implies $O'I$ is the Euler line of $\triangle DEF$.Therefore,$OI$ is the Euler line of $\triangle DEF$, as desired.