A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $${60^ \circ }$$ and $${30^ \circ }$$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $$P$$ and $$Q$$ respectively on its path. Let $$\theta $$ be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between $$P$$ and $$Q$$. Find the numerical value of $${\tan ^2}\theta $$. (Assume that the observer is not inside the vertical projection of the path of the bird.)

Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3 $$.

A tower $$AB$$ leans towards west making an angle $$\alpha $$ with the vertical. The angular elevation of $$B$$, the topmost point of the tower is $$\beta $$ as observed from a point $$C$$ due west of $$A$$ at a distance $$d$$ from $$A$$. If the angular elevation of $$B$$ from a point $$D$$ due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma $$, then prove that $$2$$ tan $$\alpha = - \cot \beta + \cot \gamma $$.

Let $${A_1},{A_2},........,{A_n}$$ be the vertices of an $$n$$-sided regular polygon such that $${1 \over {{A_1}{A_2}}} = {1 \over {{A_1}{A_3}}} + {1 \over {{A_1}{A_4}}}$$, Find the value of $$n$$.