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(a) Suppose p and q are points on the unit circle such that the line through p and q intersects the real axis. Prove that if z is the point where this line intersects the real axis, then

$z=\frac{p+q}{pq+1}$

(b) Let $P_1P_2...P_{18}$ be a regular 18-gon. Prove that $P_1P_{10}$, $P_2P_{13}$, and $P_3P_{15}$ are concurrent.

As for (a), I knew that z must be real given that it is on the real axis. Also, I can use the collinearity equation $\frac{v-z}{u-z} = \overline{\left(\frac{v-z}{u-z}\right)}$ and plug in values p, q, and z. Given that z is real, $\overline{z}=z$ so our equation for collinearity is now $\frac{v-z}{u-z}=\frac{\overline{v}-z}{\overline{u}-z}$. Past here I'm not really sure.

As for (b), I believe I need to use something from what is proved in (a) and therefore have not made any progress. Any help would be much appreciated!

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  • $\begingroup$ Hope I made myself clear :) $\endgroup$ – Kunal Gupta Mar 1 '15 at 17:53
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Hint:

(a) Do what you're doing, cross multiply, find $z$ and put $\bar p = \frac1p$ and $\bar q = \frac1q$, since for any complex number $\omega$

$\omega\bar\omega = |\omega|^2$

(b) $P_k$ are 18th roots of unity, use the fact that $P_k = e^{i\frac{2k\pi}{18}}$ and prove that $P_k = -P_{k+9}$ and thus lie on opposite sides of origin and pass through $z=0$.

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