6feb14 o4jun11

Problems on Euclid's Geometry

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