Iron man mark 1 (43 to go) series of challenges for geniuses

Suppose a line 'l' cuts an equilateral triangle $ABC$ of side $\sqrt{3}$ in two points different from the vertices. Say it cuts $AB$ and $AC$ in points $R$ and $Q$ respectively. We mark the orthocentre $H$ of the triangle $ARQ$ and also the midpoint $M$ of side $RQ$ . We extend $HM$ to a point $T$ so that $HM = MT$ . The point $P$ is the foot of perpendicular from $T$ on side $BC$ .

Now, we draw outward equilateral triangles $RA'Q$ , $QC'P$ , and $PB'R$ . (The Napoleonic triangles of triangle $PQR$ ).

The task is to find: $A'T + B'T + C'T$ .