If $A, B, C, D$ are the angles of a cyclic quadrilateral taken in order,then $\cos A + \cos B + \cos C + \cos D =$

The product of the distances of the incentre $I$ from the angular points $A, B, C$ of a $\Delta ABC$ is:

If $r_1, r_2$ and $r_3$ of a $\triangle ABC$ are in Harmonic progression,then $a, b$ and $c$ will be in

The angles $\alpha, \beta, \gamma$ of a triangle satisfy the equations $2 \sin \alpha + 3 \cos \beta = 3 \sqrt{2}$ and $3 \sin \beta + 2 \cos \alpha = 1$. Then,angle $\gamma$ equals (in $^{\circ}$)

With usual notations,in $\Delta ABC$,if $b \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{B}{2}=\frac{3 a}{2}$,then

Let a triangle $ABC$ be inscribed in a circle of radius $2$ units. If the $3$ bisectors of the angles $A, B$ and $C$ are extended to cut the circle at $A_1, B_1$ and $C_1$ respectively,then the value of $\left[\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}\right]^2$ is: