<s>[INST] Concurrent lines proof for a regular 18-gon 
Let $X_1 X_2 \dotsb X_{18}$ be a regular 18-gon. Show that $X_1 X_{10}$, $X_2 X_{13}$, and $X_3 X_{15}$ are concurrent.

What would be the best way to prove this? I am actually struggling understanding 'concurrent' as I am not sure how they are concurrent in a regular 18-gon
 [/INST] $\qquad\qquad$
Let $A,B,C$ be the intersection point of $X_{13}X_1$ with $X_{15}X_3$, $X_{13}X_3$ with $X_{10}X_1$, $X_{13}X_2$ with $X_1X_3$ respectively.
Then, we get
$$\begin{align}\frac{AX_1}{X_{13}A}\cdot\frac{CX_3}{X_1C}\cdot\frac{BX_{13}}{X_3B}&=\frac{X_1X_3\sin 40^\circ}{X_{13}X_3\sin 20^\circ}\cdot\frac{X_3X_{13}\sin 10^\circ}{X_1X_{13}\sin 10^\circ}\cdot\frac{X_1X_{13}\sin 30^\circ}{X_1X_3\sin 70^\circ}\\\\&=\frac{\sin 40^\circ\sin 30^\circ}{\sin 20^\circ\sin 70^\circ}\\\\&=\frac{\frac 12\sin 40^\circ}{\sin 20^\circ\sin 70^\circ}\\\\&=\frac{\sin 20^\circ\cos 20^\circ}{\sin 20^\circ \sin 70^\circ}\\\\&=1\end{align}$$
The claim follows from Ceva's theorem.
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