In a triangle $ABC$,if $(a-b)(s-c)=(b-c)(s-a)$,then $r_1+r_3=$

In $\triangle ABC$,find the value of $a^3 \cos(B-C) + b^3 \cos(C-A) + c^3 \cos(A-B)$.

Consider an obtuse-angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius $1$.
$(1)$ Let $a$ be the area of the triangle $ABC$. Then the value of $(64 a)^2$ is
$(2)$ The inradius of the triangle $ABC$ is

In a triangle $ABC$,if $\sin \frac{A}{2} = \frac{1}{4} \sqrt{\frac{3}{5}}$,$a = 2$,$c = 5$ and $b$ is an integer,then the area (in sq. units) of triangle $ABC$ is

If $A, B, C$ are the angles of a triangle,then $\sin^2 A + \sin^2 B + \sin^2 C - 2\cos A \cos B \cos C = $

With usual notations,in a triangle $ABC$,$a \cos(B - C) + b \cos(C - A) + c \cos(A - B)$ is equal to