Results 1 to 6 of 6

Math Help - Question 5

  1. #1
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Question 5

    ImPerfectHacker has requested that I set a problem this week so here it is:

    Problem of the Week 5

    Given any four distinct points A,\ B,\ C,\ O, show that the three angles between the bisectors of \angle AOB, \angle BOC and \angle COA are all acute, right or obtuse.

    RonL

    (Clarification: the bisectors are of the non-reflex angles that correspond to the specified points)
    Last edited by CaptainBlack; November 7th 2006 at 11:25 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,902
    Thanks
    329
    Awards
    1
    Quote Originally Posted by CaptainBlack View Post
    InPerfectHacker has requested that I set a problem this week so here it is:

    Problem of the Week 5

    Given any four distinct points A,\ B,\ C,\ O, show that the three angles between the
    bisectors
    of \angle AOB, \angle BOC and \angle COA are all acute, right or obtuse.

    RonL
    Perhaps I'm obtuse. What three angles?

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by topsquark View Post
    Perhaps I'm obtuse. What three angles?

    -Dan
    There are three lines of interest and they are the bisectors of \angle AOB, \angle BOC, and \angle COA respectivly.

    See the diagram where B_1,\ B_2,\ B_3 are the three bisectors.


    RonL
    Attached Thumbnails Attached Thumbnails Question 5-gash.jpg  
    Last edited by CaptainBlack; November 7th 2006 at 11:07 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by CaptainBlack View Post
    ImPerfectHacker has requested that I set a problem this week so here it is:

    Problem of the Week 5

    Given any four distinct points A,\ B,\ C,\ O, show that the three angles between the bisectors of \angle AOB, \angle BOC and \angle COA are all acute, right or obtuse.

    RonL

    (Clarification: the bisectors are of the non-reflex angles that correspond to the specified points)
    Introduce unit vectors \bold {u, v, w} along OA, OB and OC respectively. Then
    the bisectors are collinear with \bold{u+v, v+w} and \bold{u+w}. Also:

    <br />
(\bold{u}+\bold v).(\bold v+\bold w) = \bold v.\bold v + \bold u.\bold v + \bold u.\bold w + \bold v.\bold w = 1 +\bold u.\bold v + \bold u.\bold w + \bold v.\bold w<br />

    <br />
(\bold u+\bold v).(\bold u+\bold w) = \bold u.\bold u + \bold v.\bold u + \bold u.\bold w + \bold v.\bold w = 1 + \bold v.\bold u + \bold u.\bold w + \bold v.\bold w<br />

    <br />
(\bold v+\bold w).(\bold u+\bold w) = \bold w.\bold w + \bold v.\bold u + \bold w.\bold u + \bold v.\bold w = 1 + \bold v.\bold u + \bold w.\bold u + \bold v.\bold w<br />

    So these three dot products are equal, so in particular are all positive,
    zero or negative. Which implies that the angles between the bisectors
    are all acute, right or obtuse.

    RonL
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2005
    Posts
    21
    A couple of questions before I try to solve it: (The previous solution used concepts I am unfamiliar with so the problem isn't ruined for me )

    Can I assume A,B,C, and O are noncollinier?
    Am I trying to prove that all of the angles have the same quality of acute, obtuse, and/or right, or only that none of them are reflexive or lines?
    Also, how much geometry is required at mininum to find the proof? I've only taken 1 year.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by The Pondermatic View Post
    A couple of questions before I try to solve it: (The previous solution used concepts I am unfamiliar with so the problem isn't ruined for me )

    Can I assume A,B,C, and O are noncollinier?
    You can assume so, I see an exeptional and/or ambiguous cases
    if the can be colinear. What I actualy want is the O does not lie on
    any of the segmants AB, BC, AC.

    Am I trying to prove that all of the angles have the same quality of acute, obtuse, and/or right, or only that none of them are reflexive or lines?
    Also, how much geometry is required at mininum to find the proof? I've only taken 1 year.
    As I don't know a synthetic proof, I can't say.

    RonL
    Last edited by CaptainBlack; November 20th 2006 at 08:22 PM.
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum