Results 1 to 12 of 12

Math Help - regular polygon

  1. #1
    Member
    Joined
    May 2009
    Posts
    146

    regular polygon

    Choose some point A inside a regular polygon. The sum of length of line segments connecting A to the vertices of the polygon achieves minimum when A is the center of the polygon.
    Can somebody give a proof of this? Thanks!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    Is it valid? Your personal exploration would be of most benefit to you.

    Prove it for a triangle. Does it work?
    Pove it for a square. A Rhumbus. A convex quadrilateral. A concave quadrilateral. (That last one presents some LOS problems.)

    What say you? Do you believe it should be provable?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2009
    Posts
    146
    Quote Originally Posted by TKHunny View Post
    Is it valid? Your personal exploration would be of most benefit to you.

    Prove it for a triangle. Does it work?
    Pove it for a square. A Rhumbus. A convex quadrilateral. A concave quadrilateral. (That last one presents some LOS problems.)

    What say you? Do you believe it should be provable?
    Why, it should! Let's consider convex ones, say B1B2...Bn. A friend of mine told me his idea: suppose A's not the center of the polygon. Rotate the polygon by an angle of 2pi/n around it's center, then A becomes A'. So if A minimizes the sum of lines connecting to vertices, so does A'. Let this sum be L. Now choose the midpoint O of AA'. We have OBi < (ABi+A'Bi)/2. Sum both sides over i we find the sum of length connecting O to vertices is less than L. So A can't be minimizing L if it's not the center, where A=A'=O.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    Yeah, forget I said anything about concave polygons. I missed "regular".

    I like it. It not only completes the proof, but also create an algorithm (albeit infinite) for finding the center.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2009
    Posts
    146
    Quote Originally Posted by TKHunny View Post
    Yeah, forget I said anything about concave polygons. I missed "regular".

    I like it. It not only completes the proof, but also create an algorithm (albeit infinite) for finding the center.
    Indeed. And here's one without algorithm. When he first told me about the problem, my maths teacher gave a entertaining "proof": Find n long thin strings. Tie one end of them together (call the knot A). Tie the other ends each to a plumb weighing 1N. For a horizontally placed regular polygon board, let each string go through the a small hole at each vertex, with its plumb hanging below the board. Now if A's at the center O, sum of forces on it is zero so it stays there. Otherwise, sum of forces in the \vec{AO} direction is positive (using vectors it's easy to prove this!). So by moving A to O plumbs will have done work and their potential energy decreases! That means the sum of length of strings below the board will increase and the sum of length inside the board decreases! and this completes the proof.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    EDIT: late in 2 minutes.

    A question...

    Is it enough to prove the following for making the conclusion in OP's question?

    Say we have regular polygon A1A2...An with center O. Let denote \vec{OA_i}=\vec{a_i} for 1\leq  i\leq n. Then \sum_{i=1}^{n}\vec{a_i}=0
    Last edited by Also sprach Zarathustra; June 11th 2011 at 10:16 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    May 2009
    Posts
    146
    Quote Originally Posted by Also sprach Zarathustra View Post
    A question...

    Is it enough to prove the following for making the conclusion in OP's question?

    Say we have regular polygon A1A2...An with center O. Let denote \vec{OA_i}=\vec{a_i} for 1\leq  i\leq n. Then \sum_{i=1}^{n}\vec{a_i}=0
    OP's question?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    Quote Originally Posted by godelproof View Post
    OP's question?
    LOL.

    I mean to your question.



    -
    To my question, perhaps it's not enough. We talking about distances here...
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    May 2009
    Posts
    146
    N
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    May 2009
    Posts
    146
    Quote Originally Posted by TKHunny View Post
    Yeah, forget I said anything about concave polygons. I missed "regular".

    I like it. It not only completes the proof, but also create an algorithm (albeit infinite) for finding the center.
    any interesting proof you have in mind?
    for an arbitary convex polygon, i guess such point is the center of mass of the polygon (i have no idea if it's correct!). but if it's correct, then there's algorithm for finding it, too. isn't that nice!
    Last edited by godelproof; June 11th 2011 at 10:31 PM. Reason: sorry, i know the guess is incorrect!
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    May 2009
    Posts
    146
    Quote Originally Posted by Also sprach Zarathustra View Post
    LOL.

    I mean to your question.



    -
    To my question, perhaps it's not enough. We talking about distances here...
    perhaps so! summation alone won't be enough. i thing you need to multiply vectors so you can have distances and angles...
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Member
    Joined
    May 2009
    Posts
    146
    and here's the proof using vectors for Zarathustra's question, in the attachment. Interestingly, I suddenly realize that this together with the the argument in #5 also allows us to compare the sums of connections for any two points in the polygon: the one farther away from the center O has a larger sum. And the same comparison can be made even when one or both points are outside the polygon! (since there's no requirement in #5 and #12 for A to be inside the polygon)

    The problem has a nice and simple structure.

    EDIT: the contours are not circles... so we cannot compare arbitary points. The best #5 and #12 allow us is to compare 2 points if they and the center O lie on a straight line.
    Attached Thumbnails Attached Thumbnails regular polygon-proof.jpg  
    Last edited by godelproof; June 12th 2011 at 04:13 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: January 28th 2011, 02:15 PM
  2. nine-sided regular polygon
    Posted in the Geometry Forum
    Replies: 5
    Last Post: February 27th 2010, 06:05 PM
  3. Coordinate of Vertex of Regular Polygon
    Posted in the Geometry Forum
    Replies: 1
    Last Post: May 31st 2009, 04:03 PM
  4. Equation needed - Regular polygon question!
    Posted in the Geometry Forum
    Replies: 1
    Last Post: March 14th 2008, 06:01 AM
  5. Area of a Regular Polygon
    Posted in the Algebra Forum
    Replies: 5
    Last Post: August 2nd 2006, 09:31 PM

Search Tags


/mathhelpforum @mathhelpforum