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\leftline{MATH 392 -- Geometry Through History -- Quiz 1 -- February 19, 2016}
\vskip 5pt
\leftline{Your Name: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$}
\vskip 10pt
\noindent
Assume Postulates I - V of Euclid's {\it Elements} hold.  
\begin{enumerate}
\item[(A)]  (20)  Let $ABCD$ be a \emph{parallelogram} (vertices $A,B,C,D$ named counterclockwise
around the boundary) with $AB = BC = CD = AD$, so the parallelogram is 
a \emph{rhombus}.  Show that the diagonals $\overline{AC}$ and $\overline{BD}$ 
\emph{bisect each other} and meet at a right angle.  That is, show that if the diagonals 
intersect at $E$, then $AE = EC$, $BE = DE$, and $\angle AEB, \angle BEC$, etc are all right angles.  
Use only facts contained in Book I of Euclid for this.


\vfill\eject

\item[(B)] {\it Extra Credit} (10) Show that if we have the situation of Euclid's Proposition I.47
as in the figure above, then $\overline{AD}$ and $\overline{CF}$ meet at a right angle.
For this one, any complete proof is OK; don't feel constrained to argue by only Euclidean methods.
Coordinates, vectors, trig, etc. are all legal.
\begin{figure}
\begin{center}
\includegraphics[width=.35\linewidth]{../EuclidI47.jpg}
\caption{Figure for Proposition 47, Book I}
\end{center}
\end{figure}

\end{enumerate}

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