Let $I, H, O$ be the incentre, orthocentre and circumcentre of $\vartriangle ABC$ respectively. $D$ is the circumcentre of $\vartriangle AIC$. $H$ is reflected along $BC$ and $AB$ to $E$ and $F$ respectively. Prove that $D, O, F$ are collinear if and only if $DE$ is perpendicular to $EF$. by @pepemaths