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\centerline{MATH 392 -- Geometry Through History}
\centerline{Midterm Exam}
\centerline{March 22, 2016}
\vskip 10pt
\noindent
{\it Instructions:}  Do all work in the blue exam booklet.  There are 100 possible points.
\vskip 10pt
\noindent
I.  Short Answer.  Answer any four (4) of the following briefly (two or three sentences 
will suffice for each).  Only the best 4 will be used to compute your total score.
\begin{enumerate}
\item[(A)] (5) Approximately when and where was Euclid active?  How much do we
know about his life?    
\item[(B)] (5) State the 5 Postulates for geometry Euclid included at the start
of Book I of the {\it Elements}.  What was the function of these statements
in Euclid's development of geometry?
\item[(C)] (5) What was Proclus's opinion about Euclid's Postulate V?  How was this
train of thought continued in later work?
\item[(D)] (5) Approximately when and where was Girolamo Saccheri, S.J. active?  What
were his main contributions?
\item[(E)] (5) Why is Gauss now always included with Bolyai and Lobachevsky in 
discussions of the developers of hyperbolic geometry when he did not publish
any work on that subject?
\item[(F)] (5) What does Pasch's Axiom say and why do we include
that statement in a complete set of axioms for Euclidean geometry?
\end{enumerate}
\vskip 10pt
\noindent
II. (20) State and prove Proposition 29 in Book I of the {\it Elements}.  What
is especially notable about this proposition?  (Note:  If you need to, you 
may ``buy'' the statement to be proved from me for 5 of the possible points
on this problem.) 
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\noindent
III.  Definitions and related facts.
\begin{enumerate} 
\item[(A)] (10) What is the \emph{angle of parallelism} for a segment $AP$
in hyperbolic geometry?  What does the Bolyai-Lobachevsky theorem tell us
about the angle of parallelism?  
\item[(B)] (10) What is a \emph{horocycle} in hyperbolic geometry? 
What is the closest Euclidean analog to this curve?
\end{enumerate}
\vskip 10pt
\noindent
IV. Refer to the figure at the top of the next page.  A \emph{median} of a triangle is the 
line segment joining a vertex to the midpoint of the opposite side.  
In the figure, $\overline{AA'}$, $\overline{BB'}$, and the dotted line
$\overline{CC'}$ are medians.  In this problem, from the parts below you will 
find a proof of the theorem that 
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\centerline{\emph{The three medians of a triangle meet in a single point (called
the centroid of the triangle).}}
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\noindent  

\begin{figure}
\begin{center}
\includegraphics[trim=.9in 1in 1in 1in,clip,width=3in,height=3in]{Medians.png}
\caption{$\Delta ABC$ with medians $AA'$ and $BB'$ in solid black}
\end{center}
\end{figure}

\noindent
Let $O$ be the point where two medians meet as in the figure.  
The idea of this proof is to show that 
\medskip
\centerline{(*)  $O$ is a point such that $AO = 2\cdot OA'$ and $BO = 2\cdot OB'$.}
\medskip\noindent
In other words, (any) two medians \emph{cut each other} at a point $O$ in two line segments
in the ratio $1:2$, with the shorter distance being the distance from $O$
to the midpoint of the opposite side.  
\begin{enumerate}
\item[(A)] (5) Explain why establishing (*) is enough to show that the 
three medians all intersect at $O$.  (Don't think too hard about this -- it's very easy!)
\item[(B)] (5) Show that ${\rm area}(\Delta ABA') = {\rm area}(\Delta ACA')$, 
${\rm area}(\Delta OBA') = {\rm area}(\Delta OCA')$, and similarly
${\rm area}(\Delta BAB') = {\rm area}(\Delta BCB')$ and 
${\rm area}(\Delta OAB') = {\rm area}(\Delta OCB')$.
\item[(C)] (5) Deduce that 
$${\rm area}(\Delta OBA') = {\rm area}(\Delta OCA') = {\rm area}(\Delta OCB') 
= {\rm area}(\Delta OAB').$$
\item[(D)] (5) Finally, show that $AO = 2\cdot OA'$ and $BO = 2\cdot OB'$, which
is the desired statement (*) above.
\end{enumerate}
\vskip 10pt
\noindent  
V. Consider the figure on page 3.  
\begin{enumerate}
\item[(A)] (15) Show that in the hyperbolic plane
it is possible to construct three pairwise parallel
hyperbolic lines as in the figure -- that is show
how to construct three lines $\ell_1, \ell_2$, and $\ell_3$ and points $A,A'$ on
$\ell_1$, $B,B'$ on $\ell_2$ and $C,C'$ on $\ell_3$ such that  
$\overrightarrow{AA'}$ is parallel to $\overrightarrow{BB'}$
in the direction of $A'$ and $B'$, and (at the same time) 
$\overrightarrow{B'B}$ is parallel to $\overrightarrow{CC'}$ in the direction of $B$ and $C'$, 
and (at the same time) $\overrightarrow{A'A}$ is parallel to $\overrightarrow{C'C}$
in the direction of $A$ and $C$.  
\item[(B)] (5) Generalize your argument to show that there
are similar ``asymptotic $n$-gons'' of parallel lines for any $n \ge 3$.
\end{enumerate}

\begin{figure}
\begin{center}
\includegraphics[width=3in,height=3in]{HypPars.png}
\caption{An ``asymptotic triangle'' in the hyperbolic plane}
\end{center}
\end{figure}
\end{document}


{\it Format}
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The exam will cover the material on Euclidean and hyperbolic geometry that
we have studied since the start of the semester (i.e. Problem Sets 1 - 3 and
the first quiz).  There will be roughly equal coverage of the Euclidean and
hyperbolic cases.  

You should know the statements of Postulates I - V from 
Euclid, Pasch's Axiom, and definitions of terms we have used (especially
the \emph{revised} notion of parallelism and the angle of parallelism for the 
hyperbolic case, pencils of parallel lines, horocycles, horospheres.)  

Know the statement of the Bolyai-Lobachevsky theorem, and be prepared to use it 
to deduce facts about angles of parallelism in the hyperbolic case.

I will ask you to construct one or two short proofs you have not seen before (comparable 
to the proof from Quiz 1; also see the Review/Practice Problems below).  Other 
questions will ask for one or two of the more substantial assigned proofs 
listed below.  

I might also include a short answer question with several parts on aspects of the 
historical development, so you should know approximately when each of the following 
key figures lived, what their main contributions were, and how they figure into what 
we have done so far:
\begin{itemize}
\item Euclid of Alexandria
\item Proclus
\item Girolamo Saccheri, S.J.
\item Adrien-Marie Legendre
\item Janos Bolyai
\item Carl Friedrich Gauss
\item Nikolai Lobachevsky
\end{itemize} 

\vskip 10pt
\noindent
{\it Assigned Proofs}
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\begin{enumerate}
\item The proof of Proposition 29 in Book I of Euclid's {\it Elements}.  Also -- what
is especially notable about this Proposition?
\item The proof of Proposition 38 in Book I of the {\it Elements}, assuming Proposition 37.   
\item The proof of the Saccheri-Legendre Theorem (Theorem 2.2 in McCleary).  Also know
that this result holds in both Euclidean and hyperbolic geometry.  Why?
\item The proof that if $ABCD$ is a Saccheri quadrilateral on the base AB, then 
\begin{itemize} 
\item if HAA holds, then $AB < CD$ and the angle sum in any triangle is less
than 2 right angles, 
\item if HOA holds, then $AB > CD$ and the angle sum in any triangle is greater
than 2 right angles.  
\end{itemize}
(This is covered in Theorem 3.12 in McCleary.)  
\item The proof that the ratio of lengths of concentric arcs of two horocycles intercepted
by two lines in the parallel pencil is an exponential function of the distance between
the arcs.  (This is Theorem 4.9 in McCleary; we did the same proof in class as well.)
\end{enumerate}

\vskip 10pt
\noindent
{\it Review/Practice Problems}
\vskip 10pt
\begin{enumerate}
\item[(A)]  Assuming Postulate V, prove that the three internal angle bisectors of a triangle meet
in a single point.  (Where does your proof use Postulate V?)  Hint:  Take two bisectors that meet 
and join the intersection point to the third vertex.  Under what conditions does
that line bisect the angle at that third vertex?
\item[(B)]  Assuming Postulate V, prove that the three perpendicular bisectors of the sides of a 
triangle meet in a single point.  (Where did you use Postulate V?)  Hint:  You can use
the same strategy as in (A).
\item[(C)]  Look at the figure in problem 4.20 on page 70 in McCleary.  Show that in the 
hyperbolic plane, 
it is possible to construct three nonintersecting 
hyperbolic lines as in the top left diagram -- that is show
how to construct three lines $\ell_1, \ell_2$, and $\ell_3$ and points $A,A'$ on
$\ell_1$, $B,B'$ on $\ell_2$ and $C,C'$ on $\ell_3$ such that  
$\overrightarrow{AA'}$ is parallel to $\overrightarrow{BB'}$
in the direction of $A'$ and $B'$, and (at the same time) 
$\overrightarrow{B'B}$ is parallel to $\overrightarrow{CC'}$ in the direction of $B$ and $C'$, 
and (at the same time) $\overrightarrow{A'A}$ is parallel to $\overrightarrow{C'C}$
in the direction of $A$ and $C$.  Then generalize your argument to show that there
are similar ``$n$-gons'' of parallel lines for any $n \ge 3$.
\item[(D)]  Determine the derivative $\frac{d}{dx}(\Pi(x))$ and show that it is always
negative.
\item[(E)]  Let $x,y > 0$.  Express the angle of parallelism $\Pi(x+y)$ 
in terms of $\Pi(x)$ and $\Pi(y)$.
\end{enumerate}
\end{document}
