The set of all real numbers $$a$$ such that $${a^2} + 2a,2a + 3$$ and $${a^2} + 3a + 8$$ are the sides of a triangle is ...........

In a triangle $$ABC$$, if cot $$A$$, cot $$B$$, cot $$C$$ are in A.P., then $${a^2},{b^2},{c^2}$$, are in ............... progression.

$$ABC$$ is a triangle with $$\angle B$$ greater than $$\angle C.\,D$$ and $$E$$ are points on $$BC$$ such that $$AD$$ is perpendicular to $$BC$$ and $$AE$$ is the bisector of angle $$A$$. Complete the relation $$$\angle DAE = {1 \over 2}\left[ {\left( {} \right) - \angle C} \right]$$$

$$ABC$$ is a triangle, $$P$$ is a point on $$AB$$, and $$Q$$ is point on $$AC$$ such that $$\angle AQP = \angle ABC$$. Complete the relation $$${{area\,\,of\,\,\Delta APQ} \over {area\,\,of\,\,\Delta ABC}} = {{\left( {...} \right)} \over {A{C^2}}}$$$