6feb14
o4jun11
Problems on Euclid's Geometry 



\textit{$\C$  2011, 2015 Prof. George K. Francis, Mathematics Department,
University of Illinois} 

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\maketitle


\section{Introduction}
On this page we review strategies for proving the various equivalences
between Euclid's Fifth Postulate (E5), Playfair's Axiom (PLAYF), and 
others. Although equivalence is transitive, we ask you to resist 
applying it in specific problems. In other words, even though we proved
that $E5 \equiv PLAYF) in an earlier lesson, you should not relie on 
this equivelence in proving other equivalences, for example that 
Euclid's Proposition 20 (see below) is equivalent to Playfair. 

Also, remember that an equivalence $A \equiv B$ consist in two 
different theorems: $ A \Rightarrow B$ and $B \Rightarrow A$. In 
rare instances, one can design the proof so that the equivalence 
can be shown in its entirety in a single step. You should avoid this
format while you are learning this material.

Since all of the equivalent axioms have the form $H \implies C$, read
\textit{if Hypothesis then Conclusion}, the logical form of the 
theorem you will prove have
the logical form \[ (H1\implies C1) \implies (H2 \implies C2) \]. 
But in fact, you should prove them by the
equivalent strategy \[ H2 \wedge  (H1\implies C1) \implies  C2 ,\]. 
That is we add the hypothesis of the second implication to the 
full hypothesis.  

Finally, it is a bad idea to unroll the hypothesis $H1 \implied C1$
first. First explore the hypothesis $H2$ and see how far, given all
the theorems of absolute geometry, you can get towards the conclusion
$C2$ you are seeking to verify. Then, at the appropriate moment, invoke
the implication in the hypothesis.

Playfair figures in all of the problems. Recall that we stated it
in two ways, one with the "there exists a unique parallel" and 
the second unrolls the uniqueness into a longer form. This is to 
remind you that when you prove that $ A \implies PLAYF $ you must 
show that there exists a parallel (which is easy once you remember
that Euclid's parallels are part of absolute geometry). It is the 
uniqueness that needs to be proved.

Recall the propositions in question in Lesson E2. In

\begin{itemize}
\item \textbf{PLAYF: Playfair} $ \neg(Pm) \implies (\exists ! h)(hP \and h \parallel m )$
\item \textbf{PPP: Perp. Proclus} $  (h \parallel k \and  t \perp h \and (tk)) \im
plies t \perp k$
\item \textbf{PROC:  Proclus} $ (h \parallel k \  \and \   (th) ) \implies (tk) $.
\item \textbf{TRANS: Euclid 30} $ (h \parallel k \  \and \ k \parallel \ell) \implies (h = \ell \or h \parallel \ell) $.
\item \textbf{EXAT: Euclid 32} The exterior angle of a triangle is the sum of the opposite interior angles.
\end{itemize}

 Problem Set 

\begin{itemize}
\item{Problem 0.} Prove that $PPP \equiv PLAYF$ 
\item{Problem 1.} Prove that $PROC \equiv PLAYF$ 
\item{Problem 2.} Prove that $TRANS \equiv PLAYF$ 
\item{Problem 3.} Prove that $PPP \equiv PLAYF$ 
\item{Problem 4.} Prove that $EXAT \equiv PLAYF$ 
\end{itemize}
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