Mathematics - Properties of Triangle Question with Solution | TestHub

$$\frac{a}{3\sin x}=\frac{AD}{\sin B}..\dots \dots \dots \dots \dots \left(1\right)$$

$$\begin{array}{rlr}& & \frac{2a}{3\sin (x+y)}=\frac{AE}{\sin B}\end{array}..\dots \dots \dots \dots \left(2\right)$$

$$\frac{(1)}{(2)}\Rightarrow \frac{\sin (x+y)}{2\sin x}=\frac{AD}{AE}..\dots \dots \dots \dots \left(3\right)$$

$$\frac{2a}{3\sin (y+z)}=\frac{AD}{\sin C}\dots \dots \dots \dots \left(4\right)$$

$$\frac{a}{3\sin z}=\frac{AE}{\sin C}.\dots \dots \dots \dots \left(5\right)$$

$$\frac{(5)}{(4)}\Rightarrow \frac{\sin (y+z)}{2\sin z}=\frac{AE}{AD}.\dots \dots \dots \dots \left(6\right)$$

$$\therefore (3)\times (6)\Rightarrow \frac{\sin (x+y)\sin (y+z)}{\sin x\cdot \sin z}=4$$