In a triangle $ABC$ with usual notations,if $3a = b + c$,then $\cot \frac{B}{2} \cdot \cot \frac{C}{2} =$

If in $\triangle ABC$,with usual notations,the angles are in $A$.$P$.,then $\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A =$

In a triangle $ABC$,with usual notations,if $\frac{b+c}{11} = \frac{c+a}{12} = \frac{a+b}{13}$,then $\cos A : \cos B : \cos C$ is

In $\triangle ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then

In $\triangle ABC$,if $\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}$ with usual notations,then the triangle is

In a triangle $PQR$,let $\angle PQR = 30^{\circ}$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and $10$,respectively. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ $\angle QPR = 45^{\circ}$
$(B)$ The area of the triangle $PQR$ is $25\sqrt{3}$ and $\angle QRP = 120^{\circ}$
$(C)$ The radius of the incircle of the triangle $PQR$ is $10\sqrt{3} - 15$
$(D)$ The area of the circumcircle of the triangle $PQR$ is $100\pi$