Trigonometry · Superchallenge problem

Points $A$ and $B$ are $1000$ m apart on level ground. A mountain peak $T$ stands directly above a point $H$ on the same level ground. Surveyors measure the horizontal angles $\angle HAB = 30^\circ$ and $\angle HBA = 105^\circ$, and from $B$ the angle of elevation of the peak is $60^\circ$. Find the height $TH$ of the peak above the ground, in metres, in simplest radical form.