Can you provide a proof for the following proposition:

Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , respectively. Let $A'$ be a reflection of the point $A$ with respect to the point $E$ , $B'$ reflection of the point $B$ with respect to the point $F$ and $C'$ reflection of the point $C$ with respect to the point $D$. Consider the three circles $k_1,k_2,k_3$ defined by the points $AHA'$ , $BHB'$ and $CHC$' , respectively. I claim that $k_1$,$k_2$ and $k_3$ meet at a common point $P$.

GeoGebra applet that demonstrates this proposition can be found here.