If the reciprocals of the lengths of the sides of a $\triangle ABC$ are in harmonic progression,then its ex-radii $r_1, r_2, r_3$ are in

In $\triangle ABC$, with usual notations, if the angles are in Arithmetic Progression and $3c^2 = 2b^2$, then the measure of angle $A$ is

If in $\Delta ABC$,$2b^2 = a^2 + c^2$,then $\frac{\sin 3B}{\sin B} = $

In a triangle $ABC$ with a fixed base $BC$,the vertex $A$ moves such that $\cos B + \cos C = 4 \sin^2 \frac{A}{2}$. If $a, b,$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B,$ and $C$,respectively,then:
$(A) b+c=4a$
$(B) b+c=2a$
$(C) \text{locus of point } A \text{ is an ellipse}$
$(D) \text{locus of point } A \text{ is a pair of straight lines}$

In $\triangle PQR$,if $\angle R = \frac{\pi}{4}$ and $\tan(\frac{P}{3})$,$\tan(\frac{Q}{3})$ are the roots of the equation $ax^2 + bx + c = 0$,then:

In $\triangle ABC$,if $\tan A + \tan B + \tan C = 6$ and $\tan A \cdot \tan B = 2$,then $\tan C = \dots$