question_answer

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and \[2s=x+y+z\]. If \[\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}\] and area of in circle of the triangle XYZ is \[\frac{8\pi }{3},\] then

A)  Area of the triangle XYZ is \[6\sqrt{6}\]

B)  The radius of circumcircle of the triangle XYZ is \[\frac{35}{4}\sqrt{6}\]

C)  \[\sin \,\,\frac{X}{2}\,\,\sin \,\,\frac{Y}{2}\,\,\sin \,\,\frac{Z}{2}=\,\,\frac{4}{35}\]

D)  \[{{\sin }^{2}}\,\,\left( \frac{X+Y}{2} \right)=\frac{3}{5}\]

E)  All of these