Trigo!
Let the circumradius be $1$ and:
$X = A_5 A_{11} \cap A_1 A_8$
$Y = A_5 A_{11} \cap A_6 A_9$
$Z = A_5 A_{11} \cap A_1 A_3$
We need to prove $Y = Z$.
Or $XY = XZ$
$$1+ \cos \frac {3 \pi}{7} = 2 \sin ^2 \frac {2 \pi}{7}$$$$\cot \frac {\pi }{7} \bigg(2 \sin \frac {2\pi}{7} \cos \frac {2\pi}{7} \bigg) = \cos \frac {3 \pi}{7}$$$$2 \cos \frac {2 \pi }{7} \cos \frac {3 \pi }{7} = \cos \frac {\pi}{7} - \cos \frac {2\pi}{7}$$$$2 \sin \frac {2\pi}{7} \cos \frac {3\pi}{7} = \sin \frac {2\pi}{7} - \sin \frac {\pi}{7} $$
(in order) to get :
$$\cot \frac {\pi}{7} = \frac {1 + \cos \frac {2 \pi}{7}}{\sin \frac {2 \pi}{7}}$$Which is true