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.
You are watching: The region of a plane inside of an angle
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.
edited Nov 29 "13 at 12:07
request Nov 17 "13 in ~ 14:30
2,82011 yellow badge2323 silver badges4343 bronze badges
| present 13 an ext comments
2 answers 2
energetic oldest Votes
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,sinoldsymbol R_+cup \$. Have the right to you walk from here?
edited Nov 17 "13 at 16:59
answered Nov 17 "13 in ~ 15:06
Michael HoppeMichael Hoppe
15.9k33 gold badges2727 silver badges4444 bronze badges
| show 6 much more comments
Interesting trouble ... Not sure if this is totally correct however I tried using the plane separation axiom (PSA) twice.
See more: What State Has A Capital Named For The Seventh President Of The United States?
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.