### Solution of Triangle

goals
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: