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 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)

Excellent feedback for after 11pm! I've changed the page so the error in Line 5 is now corrected. The rays are now called $\mathcal L = \left\{ {q + s \mathbf u: s \in \R_{\ge 0} }\right\}$, and $\mathcal L' = \left\{ {q + s' \left({ \mathbf u }\right) : s' \in \R_{\ge 0} }\right\}$. As the variables has been changed from $r, r'$ to $s, s'$, this should also clear things up in Line 13.

• Line 3: Good call. Before I put in the link, I was looking at the general page Definition:Adjacent which states that "The two sides of a triangle that form a particular vertex are adjacent to that vertex". Obviously, Definition:Adjacent (in Triangle) states something different. So I would guess that Definition:Adjacent (in Triangle) needs to be updated, so it mentions vertices as well as angles. Am i right?

• Line 10: True. A simple proof that two non-parallel lines in $\R^2$ only intersects in one point would suffice, but then, I don't think we have a proper definition of a line in $\R^2$, we only have geometry based on Euclidean axioms. It would be nice if a geometer (I'm not one) would upload some proofs that show how the Euclid's definitions look like in general vector spaces. I'm uploading this theorem because I need it for the Complex Analysis sections. Since "simple" stuff like this is invariably hand-waved in analysis textbooks, I'm forced to come up with my own proofs. --Anghel (talk) 13:04, 4 March 2013 (UTC)

Agreed on updating the adjacency definition; it'd be useful terminology.

Regarding a modern treatment of classical geometry; I've added some elementary definitions for geometry in affine (i.e. vector) spaces, but I don't have much time to put up more, and it's a way off the material needed for this stuff (as you mention, no definition of a line yet). --Linus44 (talk) 14:30, 4 March 2013 (UTC)

The stuff may safely be postponed until a source and someone knowledgeable and willing enough to cover it come along; e.g. stub pages with adequate maintenance tags could be created if it is obvious that something should exist as a separate result. I'm definitely not that person - I consider my time more valuably contributed in the logic and abstract algebra (notably category theory) departments. — Lord_Farin (talk) 16:38, 4 March 2013 (UTC)