BALKAN-SL 2014 G7

Let $I$ be the incenter of $\triangle ABC$ and let $H_a$ , $H_b$ , and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$ , and $\triangle AIB$ , respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$ . If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$ , prove that the line $H_aT$ is perpendicular to $BC$