Solution of Triangle

Prove the following :
$$\sqrt{rr_1r_2r_3}$$ = S
$$OA$$ and $$OB$$ are equal sides of an isosceles triangle lying in the first quadrant, where $$O$$ is the origin. $$OA$$ and $$OB$$ make angles $${ \Psi  }_{ 1 }$$ and$${ \Psi  }_{ 2 }$$ respectively with the +ive axis. Show that the slope of the bisector of the acute angle $$AOB$$ is $$co\sec { \Psi  } -\cot { \Psi  } $$ where $$\Psi ={ \Psi  }_{ 1 }+{ \Psi  }_{ 2 }\quad $$
If in a $$\triangle ABC,{a}^{2}+{b}^{2}+{c}^{2}=8{R}^{2},$$ where $$R=$$ circumradius,then the triangle is
$$AL,BM$$ and $$CN$$ are perpendiculars from angular points of a triangle $$ABC$$ on the opposite sides $$BC,CA$$ and $$AB$$ respectively.$$\triangle$$ is the areas of triangle $$ABC, r$$ and $$R$$ are the inradius and circumradius .
On the basis of the above information,answer the following questions:
In a triangle $$ABC, \angle{A}=\dfrac{\pi}{3},b=40,c=30,AD$$ is the median through $$A,$$ then $$4{\left(AD\right)}^{2}$$ must be: