BALTIC-WAY 2023 Problem 13

Let $A B C$ be an acute triangle with $A B<A C$ and incentre $I$. Let $D$ be the projection of $I$ onto $B C$. Let $H$ be the orthocentre of $A B C$. Given $\angle I D H=\angle C B A-\angle A C B$, prove that $A H=2 \cdot I D$.