Talk:Characterization of Interior of Triangle

I started reading, the proof as far as I've seems good. Some (minor) notes on the first few lines under the heading "Interior is subset" (these may or may not be useful, as I'm breaking the general don't do mathematics after 11pm rule...):


 * Line 3: "Let $S_i$ be the side of $\triangle$ that is adjacent to $A_i$ and $A_j$..." the link to Definition:Adjacent (in a Triangle) says that sides must be adjacent to an angle. Since $A_i$ and $A_j$ are distinct vertices of the triangle, I think this should read  "Let $S_i$ be the side of $\triangle$ that passes through $A_i$ and $A_j$"
 * Line 5: "Define two rays $\mathcal L = \left\{ {q + r \mathbf u: r \in \R_{\ge 0} }\right\}$, and $\mathcal L' = \left\{ {q + r' \mathbf u: r' \in \R_{\ge 0} }\right\}$. Here the point $q$ and vector $\mathbf u$ are the same in the definition of both rays. You have only changed the parameter $r$ to $r'$. Therefore the rays $\mathcal L$ and $\mathcal L'$ are equal for all possible $\mathbf u$ and $q$. I think you mean to set $\mathcal L' = \left\{ {q + r' \mathbf u: r' \in \R_{\le 0} }\right\}$
 * Line 10: "As $\mathcal L \cup \mathcal L'$ is a straight line, and $\mathcal L \cap \mathcal L' = \left\{ {q}\right\}$, it follows that $\mathcal L$ and $\mathcal L'$ cannot both intersect the same side." Agreed on this, but possibly should be linked to a (probably non-existent) proof of this fact?
 * Line 13: "where either $r, -r' \in \R_{>0}$ or $-r, r' \in \R_{>0}$"...the correctness of this depends on whether you mean $r,r'$ to be the parameters pertaining to $\mathcal L$, $\mathcal L'$ respectively, or you allow the possibility that $r$ is the parameter for $\mathcal L'$ and $r'$ the parameter for $\mathcal L$. If the former is true, according to the present definition we have $r,r' \geq 0$, so both of these are impossible, and if in the def of $\mathcal L$, $\mathcal L'$ it is supposed to read $r \in \R_{\ge 0}$, $r' \in \R_{\le 0}$ then we will always have $r, -r' \in \R_{>0}$, so the possibility $-r, r' \in \R_{>0}$ need not be considered. If the latter is true, then obviously it's fine, but a change of notation for $r,r'$ in this line might clear things up.

Final note: the fact that the line segments $\mathcal L$, $\mathcal L'$ are not contained in $S_j$ is used implicitly (the relevant result must be somewhere on this site...) --Linus44 (talk) 00:16, 4 March 2013 (UTC)