Consider the following statements connecting a triangle $$ABC$$

(i) The sides $$a, b, c$$ and area $$\Delta $$ are rational.

(ii) $$a,\tan {B \over 2},\tan {c \over 2}$$ are rational.

(iii) $$a,\sin A,\sin B,\sin C$$ are rational.
Prove that $$\left( i \right) \Rightarrow \left( {ii} \right) \Rightarrow \left( {iii} \right) \Rightarrow \left( i \right)$$

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 $$n$$.

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

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