A circle $\mathcal{O}$ is centered at point $O$, and there's a line $d$ which doesn't have common point with $\mathcal{O}$. Let point $H$ be the projection of $O$ on $d$.
Consider a point $M$ that moves along $d$. Let $\overline{MA}$ and $\overline{MB}$ tangent with $\mathcal{O}$ at points $A$ and $B$ respectively. The points $K$ and $I$ are the projections of $H$ on $\overleftrightarrow{MA}$ and $\overleftrightarrow{MB}$ respectively.
Prove that when $M$ moves along $d$, the extended line $\overleftrightarrow{KI}$ passes through a fixed point.
I think the intersection of $\overleftrightarrow{KI}$ and $\overline{OH}$ is the fixed point, but I do not know how to prove it. Any ideas?
