Question

.3. Let A and B be distinct points. Prove that for each real number r E (-00, oo) there is exactly one point on the extended line AB such that AX/XB- r. Which point on AB does not correspond to any real number r? 4. Draw an example of a triangle in the extended Euclidean plane that has one ideal vertex. Is there a triangle in the extended plane that has two ideal vertices? Could there be a triangle with three ideal vertices?

Answer 1

$\textup{If }r\neq -1, \textup{ choose }s=1-\frac{1}{r+1} ,\textup{ where }s=\textbf{AX}/\textbf{AB }.$ $\therefore s(\textbf{AB})=\textup{\textbf{AX}}.\, \textup{Since }\textbf{AX+XB=AB}, \textup{ we get }\textbf{XB=}(1-s)\textbf{AB}.\, \, \therefore \textbf{AX/XB}=s/(1-s)$ $\therefore \frac{s}{1-s}=\frac{1-\frac{1}{r+1}}{\frac{1}{r+1}}=r+1-1=r$ $\textup{If } \textbf{X }\neq \textbf{Y},\, \, \textup{then }\textbf{\textbf{AY}}/\textbf{YB}= -1-\frac{1}{1-t}.\,\, t=\textbf{AY}/\textbf{AB}.$ \

$\textup{Since }t\neq s,\,\,\textbf{AY/YB}\neq r.$ $\textup{If } \textbf{X=B}, \textup{ then }\textbf{AX/XB}\textup{ is undefined}$