If $\tan \frac{\pi }{9},x,\tan \frac{5\pi }{18}$  are in A.P and $\tan \frac{\pi }{9},y,\tan \frac{7\pi }{18}$  are also in A.P. then

$2x=Tan\frac{\pi }{9}+Tan\frac{5\pi }{18}\Rightarrow 2x=\tan 20+\tan 50=\frac{\sin 70}{\cos 20\cos 50}=\sec {50}^0$

$2y=\tan \frac{\pi }{9}+\tan \frac{7\pi }{18}\Rightarrow 2y=\tan {20}^0+\tan {70}^0$

$=2\cos ec40=2\sec {50}^0$

$\therefore y=2x$