Trigonometry · Superchallenge problem

A vertical tower stands on level ground. From a point $P$ on the ground $16$ m from the base of the tower, the angle of elevation of its top is $\theta$. From a point $R$ on the ground $4$ m from the base, the angle of elevation of the top is $90^\circ - \theta$. From a third point $Q$ on the ground, the angle of elevation of the top is $2\theta$. How far is $Q$ from the base of the tower, in metres?