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\begin{document}
\title{Math 228: Homework 8}
\author{Mary Radcliffe}
\date{due 21 Nov 2016}
\maketitle

Complete the following problems. Fully justify each response. You need only turn in those problems marked with a (*).
\begin{enumerate}

\item (*) Suppose we have $n$ lines in the plane in general position, and then $k$ other lines that are all parallel to each other, but not parallel to any of the original $n$ lines. How many regions will the plane be divided into if all $n+k$ lines are drawn? Prove that your answer is correct.

\item (*) Complete problem 11.3.2 on page 186.

\item Complete problem 11.3.7 on page 187.

\item (*) Complete problem 11.3.8 on page 187.


\item (*) Let $P$ be a convex hexagon. Suppose you triangulate $P$ by dividing $P$ into some triangles using diagonals (see figure). In how many ways can this be done?

\vspace{-1.2in}
\begin{figure}[htp]
\hspace{-.8in}
\includegraphics[scale=.6]{triangulation.pdf}
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\caption{A triangulation of a hexagon.}
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\item Let $P$ be a convex $n$-gon. Suppose you triangulate $P$ by dividing $P$ into some triangles using diagonals (see figure). In how many ways can this be done?


\item (*) For any $n$, define $H(n)$ to be the smallest number such that any set of $H(n)$ points in general position in the plane contains a convex $n$-gon, and moreover that convex $n$-gon contains no other points from the set in its interior. (This is sometimes called an {\it empty} polygon.)

Prove that $H(3)=3$ and $H(4)=5$.

(Note: It's also known that $H(5)=10$, and that $H(6)$ is finite. $H(7)$ is known to be infinite; that is, you cannot guarantee an empty convex 7-gon no matter how many points you include. But $H(6)$ is still unknown.)

\end{enumerate}
\end{document}