In a triangle $ABC$ with usual notations,if $\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}$,then the triangle is equilateral. If the side length is $a = \sqrt{6}$,find the area of the triangle.

Consider a triangle $PQR$ having sides of lengths $p, q$ and $r$ opposite to the angles $P, Q$ and $R$,respectively. Then which of the following statements is (are) $TRUE$?
$(A)$ $\cos P \geq 1-\frac{p^2}{2qr}$
$(B)$ $\cos R \geq \left(\frac{q-r}{p+q}\right) \cos P + \left(\frac{p-r}{p+q}\right) \cos Q$
$(C)$ $\frac{q+r}{p} < 2 \frac{\sqrt{\sin Q \sin R}}{\sin P}$
$(D)$ If $p < q$ and $p < r$,then $\cos Q > \frac{p}{r}$ and $\cos R > \frac{p}{q}$

With usual notations in a triangle $ABC$,the product $(I I_1) \cdot (I I_2) \cdot (I I_3)$ has the value equal to

In a triangle $ABC$,if $A = \frac{\pi}{4}$ and $\tan B \tan C = K$,then $K$ must satisfy:

In a triangle $ABC$,evaluate the expression $\frac{a(r_1 r + r_2 r_3)}{r_1 - r + r_2 + r_3}$.

In a $\Delta ABC$,the sides $a$,$b$,$c$ are the roots of the equation $x^3 - 11x^2 + 38x - 40 = 0$; then $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = $