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\title{Homework 1: due Sept 12}
\author{MATH 300, Jonathan Schneider}
\date{Fall 2018}
\maketitle




\section{Your assignment}

For this homework assigment, you must submit a proof of the Pythagorean Theorem, different from the one given below.  The theorem and proof should be clearly explained and illustrated.  Submit the completed document as a .pdf; also submit the .tex source file.  Homework should be submitted thru uic.blackboard.com.\\
\\
You will need a LaTeX typsetter appropriate for your computer system.  For Windows, I recommend TexWorks.  For Macintosh, I recommend TexShop.  You may create your document by modifying the .tex source file for this prompt.\\
\\
For guidance on how to use LaTeX, you may consult
\begin{itemize}
\item the internet
\item human beings such as myself, Jonathan Schneider
\item the source file for this prompt.
\end{itemize}

Note that any graphics you wish to include must be saved in the same location as the source file.\footnote{Alternatively but not recommended, you may specify a path in the includegraphics command.}  It is not necessary to submit the image files with your completed assignment.

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\section{Example}

The following proof is informal; you may make yours as rigorous or as glib as you please.  I really only care that you're using LaTeX.

\begin{figure}[H]\centering
\includegraphics[width=1.5in]{triangle}
\caption{}\label{triangle}
\end{figure}\noindent

\begin{theorem}
A right triangle in the plane, with sides lengths $a,b,$ and $c$ as shown in figure \ref{triangle}, satisfies the equation
\[ a^2+b^2=c^2.\]
\end{theorem}

\begin{proof}
Four wooden copies of the triangle are placed inside the corners of a square frame of dimensions $(a+b)\times(a+b)$, as shown on the left of figure \ref{PyProof}.  There is an unoccupied square of dimensions $c\times c$ in the center; its area is $c^2$.\\
 \begin{figure}[H]\centering
\includegraphics[width=3.5in]{PyProof}
\caption{}\label{PyProof}
\end{figure}\noindent
The wooden triangles may be slid through the empty space as indicated.  The resulting configuration features two unoccupied square spaces of dimensions $a\times a$ and $b\times b$, respectively; its total area equals $a^2+b^2$.  Since the area of unoccupied space is conserved during rearrangement, the result follows.
\end{proof}




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