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.

A vertical tower $$PQ$$ stands at a point $$P$$. Points $$A$$ and $$B$$ are located to the South and East of $$P$$ respectively. $$M$$ is the mid point of $$AB$$. $$PAM$$ is an equilateral triangle; and $$N$$ is the foot of the perpendicular from $$P$$ and $$AB$$. Let $$AN$$$$=20$$ mrtres and the angle of elevation of the top of the tower at $$N$$ is $${\tan ^{ - 1}}\left( 2 \right)$$. Determine the height of the tower and the angles of elevation of the top of the tower at $$A$$ and $$B$$.

$$ABC$$ is a triangular park with $$AB=AC=100$$ $$m$$. A television tower stands at the midpoint of $$BC$$. The angles of elevetion of the top of the tower at $$A, B, C$$ are 45$$^ \circ $$, 60$$^ \circ $$, 60$$^ \circ $$, respectively. Find the height of the tower.

A sign -post in the form of an isosceles triangle $$ABC$$ is mounted on a pole of height $$h$$ fixed to the ground. The base $$BC$$ of the triangle is parallel to the ground. A man standing on the ground at a distance $$d$$ from the sign-post finds that the top vertex $$A$$ of the triangle subtends an angle $$\beta $$ and either of the other two vertices subtends the same angle $$\alpha $$ at his feet. Find the area of the triangle.