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 $$\alpha = - \cot \beta + \cot \gamma $$.

An observer at $$O$$ notices that the angle of elevation of the top of a tower is $${30^ \circ }$$. The line joining $$O$$ to the base of the tower makes an angle of $${\tan ^{ - 1}}\left( {1/\sqrt 2 } \right)$$ with the North and is inclined Eastwards. The observer travels a distance of $$300$$ meters towards the North to a point A and finds the tower to his East. The angle of elevation of the top of the tower at $$A$$ is $$\phi $$, Find $$\phi $$ and the height of the tower.

Three circles touch the one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $$4$$. Find the ratio of the product of the radii to the sum of the radii of the circles.

A man notices two objects in a straight line due west. After walking a distance $$c$$ due north he observes that the objects subtend an angle $$\alpha $$ at his eye; and, after walking a further distance $$2c$$ due north, an angle $$\beta $$. Show that the distance between the objects is $${{8c} \over {3\cot \beta - \cot \alpha }}$$; the height of the man is being ignored.