ns am examining Euclidian geometry and also I i found it that any kind of angle divides a plane into 2 regions: an inside and an outside. Is over there a require for a proof of this (something follow me the present of Jordan theorem), or is it simply "obvious"?

Browsing the internet, ns came throughout a following simpler version: any type of line divides a airplane into 2 regions. Perhaps someone will find it relevant.

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My present understanding is the the PSA dram a crucial role in this type of thing. Uneven you space doing the the analytic geometry way, in which instance the PSA must be somehow already "coded in" ...I think.

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edited Nov 29 "13 at 12:07
request Nov 17 "13 in ~ 14:30 $endgroup$
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Let $vec v$ (the vertex), $vec a$, $vec b$ (the rays) that an edge in $oldsymbol R^2$. The internal $I$ that the angle might be defined as $I:=\vec v+tcdotvec a+scdotvec bmid t,sinoldsymbol R_+cup\$. Have the right to you walk from here?

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edited Nov 17 "13 at 16:59
answered Nov 17 "13 in ~ 15:06 Michael HoppeMichael Hoppe
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Interesting trouble ... Not sure if this is totally correct however I tried using the plane separation axiom (PSA) twice.

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Between: unknown (along with point, line, on, and congruent, cf. Hilbert).

Same side: permit $itl$ it is in a line and let A and also B be two points which room not top top $itl.$ clues A and B space $extiton the same side$ of $itl$ if either $itl$ and also $leftarrow abdominal muscle ightarrow$ carry out not intersect at all, or if they carry out intersect yet the allude of intersection is not between A and B.

PSA: For any type of line $itl$ and also points $A, B, C$ which room not on $itl:$ (i) if A and also B space on the same side that $itl$ and A and C are on the same side the $itl,$ climate B and also C space on the same side the $itl;$ (ii) If A and B room not on the exact same side that $itl$ and also A and C are not ~ above the same side of $itl,$ then B and C are on the exact same side of $itl.$

The meaning of angle inner is

A allude lies in the interior or is an interior component of $angle BAC$ if it is ~ above the same side of abdominal as C and also the exact same side that AC together B.

(ref: http://www.mcs.uvawise.edu/msh3e/resources/geometryBook/geometryBook.html)