If the altitudes of a triangle are in arithmetic progression,then the sides of the triangle are in

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 $100\pi$

In $\triangle ABC$,if $a=5, b=4$ and $\cos(A-B)=\frac{31}{32}$,then $c=$

In a triangle,if the angles are in the ratio $3: 2: 1$,then the ratio of its sides is

The number of triangles $ABC$ that can be formed with $a = 3, b = 8$ and $\sin A = \frac{5}{13}$ is

In a triangle $ABC$,if $2r_1 = 3r_2 = r_3$,then $a : b : c =$