Let $$ABC$$ be a triangle such that $$\angle ACB = {\pi \over 6}$$ and let $$a, b$$ and $$c$$ denote the lengths of the sides opposite to $$A$$, $$B$$ and $$C$$ respectively. The value(s) of $$x$$ for which $$a = {x^2} + x + 1,\,\,\,b = {x^2} - 1\,\,\,$$ and $$c = 2x + 1$$ is (are)

Let $$ABCD$$ be a quadrilateral with area $$18$$, with side $$AB$$ parallel to the side $$CD$$ and $$2AB=CD$$. Let $$AD$$ be perpendicular to $$AB$$ and $$CD$$. If a circle is drawn inside the quadrilateral $$ABCD$$ touching all the sides, then its radius is

One angle of an isosceles $$\Delta $$ is $${120^ \circ }$$ and radius of its incircle $$ = \sqrt 3 $$. Then the area of the triangle in sq. units is

In a triangle $$ABC$$, $$a,b,c$$ are the lengths of its sides and $$A,B,C$$ are the angles of triangle $$ABC$$. The correct relation is given by