Let $P_1 P_2 \dotsb P_{18}$ be a regular 18-gon. Show that $P_1 P_{10}$, $P_2 P_{13}$, and $P_3 P_{15}$ are concurrent.
When positioned on the unit circle, I know that $P_1 P_{10}$ is the diameter. I also know that if $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis and if $z$ is the point where this line intersects the real axis, then $z = \dfrac{p+q}{pq+1}$. So I should let $P_1 P_{10} be on the real axis. Where should I go from now? Thanks.