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

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 $$\angle B = $$.................

If in a triangle $$ABC$$, $${{2\cos A} \over a} + {{\cos B} \over b} + {{2\cos C} \over c} = {a \over {bc}} + {b \over {ca}},$$ then the value of the angle $$A$$ is .................... degrees.

If the angles of a triangle are $${30^ \circ }$$ and $${45^ \circ }$$ and the included side is $$\left( {\sqrt 3 + 1} \right)$$ cms, then the area of the triangle is ...............