Class Notes on the Proofs of Ceva and Menelaus

\begin{document} \maketitle This lesson presents the class work in texPad of the four theorems that establish both Ceva and Menelaus' theorem. \subsection{Logical form of the Proofs.} Each, Ceva and Menelaus, has the logical form $ P \implies (Q \sim R)$, where $P,Q,R$ are three propositions. The equivalence, in turn, means that $ (Q \implies R) \and (R \implies Q)$. Thus there are four theorems of the form $ P \implies (Q \implies R)$. This double implication simplifies to $ (P \and Q) \implies R$. And that is how we do it. Morevover, all four have the same first assumption, namely that three points are given on the sides of a triangle. There is another, tacit assumtpion, that all of these 6 points are different from each other. This tacit assumption allows us to do some algebra without fear of dividing and through by zero, i.e. all shown fractions are real. \subsection{Technical presentation.} We decided to present the essence of each proof in real time in class. For this we used texPad. To keep the exposition short, some necessary justifications, all due to our "tacit assumption" are left to the student to insert. \subsection{The four theorems.} \subsection{Discussion.} You will learn much about proving theorems by comparing these four proofs with each other, with the proofs for the same theorems in the lessons, and even with the exposition of the proofs in Tondeur's textbook. For instance, note the similarity of the two parts for Ceva, compared to the two parts of Menelaus. Note also the difference. For Menelaus, we made the two proofs verbatim the same, until one, or the other hypothesis is made to decide the next step. Opposite conclusions are arrived a. For Ceva's theorem, the structure of the directions of the equivalence are not so similarly structured, and so you don't have to pay such close attention in keeping track of their evolution. \subsection{The LaTeX Codes.} For your convenience, here are the text files for the LaTeX code for these texPad examples. ceva-part1.txt