equilateral triangle and inscribed circle

Question

Let $ABC$ be an equilateral triangle. Let $D$ be a random point on $BC$. Let $I_1$ and $I_2$ be the incenters in $\triangle ABD$ and $\triangle ADC$. Let $O$ be the circumcentre of $\triangle AI_1I_2$. Prove that $OD$ is perpendicular to $BC$.

Answer 1

This is not an answer.

It is a (Geogebra) figure that should accompany the question.