Can you provide a proof for the following claim:
Claim.Given any $\triangle ABC$. The tangent lines to the circumscribed circle of $\triangle ABC$ are constructed at vertices $A$,$B$,$C$ . Let point $D$ be the intersection point of tangent lines that passes through vertices $A$ and $C$ , point $E$ the intersection point of tangent lines that passes through vertices $A$ and $B$ , point $F$ the intersection point of tangent lines that passes through vertices $B$ and $C$ , and let $H_1$,$H_2$,$H_3$ be the orthocenters of $\triangle ACD$, $\triangle AEB$ and $\triangle BFC$ respectively. Then line segments $AH_3$,$BH_1$ and $CH_2$ concur at the nine-point center of $\triangle ABC$.
GeoGebra applet that demonstrates this claim can be found here.
I don't know how to start the proof. All I know is that nine-point center of the triangle lies in the middle of the line segment whose endpoints are orthocenter and circumcenter , but I don't know how to use that fact. Any hints are welcomed.