In a triangle $$ABC$$, $$AD$$ is the altitude from $$A$$. Given $$b>c$$, $$\angle C = {23^ \circ }$$ and $$AD = {{abc} \over {{b^2} - {c^2}}}$$ then $$\angle B = $$.................

A circle is inscribed in an equilateral triangle of side $$a$$. The area of any square inscribed in this circle is ..............

If the lengths of the sides of triangle are $$3, 5, 7$$ then the largest angle of the triangle is

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