Let the circumcircle of $ABCD$ be the unit circle. We will apply complex numbers. Now, let $a,b,c,d,h_a,h_b,h_c,h_d$ denote $A,B,C,D,H_A,H_B,H_C,H_D$ respectively. Now,
$h_a=b+c+d$
So, the midpoint of $AH_A$ has complex coordinate $\dfrac{a+h_a}{2}=\dfrac{a+b+c+d}{2}$.
Due to the symmetric nature of the midpoints, they all share a common midpoint. So, they are concurrent.