Monday, February 28, 2011

Fox 331

Here is a simple contruction problem.
http://www.8foxes.com/

11 comments:

  1. ABCD the quadrilaterl
    E midpoint of AB;F midpoint of CD
    G midpoint of EF

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  2. No, this won't work.

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  3. the first answer works
    other solutions:
    I midpoint of AD;J midpoint of BC
    G midpoint of IJ
    or
    K midpoint of AC,L midpoint of BD
    G midpoint of KL
    and the 3 ways give the same point G!
    anonymous1

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  4. theorem
    the centroid of the vertices of a quadrilateral
    is the midpoint of the lines joining the midpoint of opposite sides,it is also the midpoint of the line joining the midpoint of the diagonals
    anonymous 1

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  5. hmmm ok! there are three possible interpretations of problem statement:
    I1) centroid of 4 equal 0-dimensional masses at quadrilateral vertices
    I2) centroid of 4 homogeneously massive 1-dimensional lines forming the given quadrilateral
    I3) centroid of a homogeneously massive 2-dimensional quadrilateral.

    @Anonymous 1: your solution solves I1 only

    I highly suspect that initial 8foxes intention was I3. Anyway, all three interpretations have a construction solution...

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  6. "I highly suspect that initial 8foxes intention was I3. Anyway, all three interpretations have a construction solution..."

    Correct. Slight coloring of the quadrilateral implies that.

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  7. I2
    the centroid of a massive line is the midpoint
    with mass proportional to the length l
    let be A',B',C',D' the midpoints of the sides a,b,c,d
    we must find K center of the point masses (A',a),(B',b)
    A'K=a/(a+b)A'B'
    this can be done with parallel lines
    L center of (C',c)and (D',d)
    G2 is the centroid of (K,a+b),(L,c+d)

    I3
    the center of gravity of an homogenious triangle is the centroid with a mass proportional to the aera
    each diagonal divides ABCD into 2 triangles
    with centroids C1,C2(first diagonal),C3,C4(second
    diagonal)
    G3 lies on the lines C1C2 and C3C4
    G3 is the intersection of this lines
    (thank you for the hints)
    anonymous1

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  8. The centroid of the vertices of a quadrilateral occurs at the point of intersection of the bimedians (i.e., the lines M(AB)-M(CD) and M(AD)-M(BC) joining pairs of opposite midpoints) (Honsberger 1995, pp. 36-37). In addition, it is the midpoint of the line M(AC)-M(BD) connecting the midpoints of the diagonals AC and BD (Honsberger 1995, pp. 39-40).

    http://mathworld.wolfram.com/Quadrilateral.html

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  9. There is a nice animation in Joe Wilson's page.
    See here:

    http://jwilson.coe.uga.edu/emt668/EMT668.Folders.F97/Patterson/EMT%20669/centroid%20of%20quad/Centroid.html

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  10. The name of the guy is Jim Wilson. Sorry about that...

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  11. this simple construction of G3
    M midpoint of diagonal BD
    O intersection of the diagonals
    on diagonal AC,AN=CO
    then MG=1/3 MN
    Essai sur la determination des centres de gravité
    Hyacinthe Celestin Gaubert (1839)p27
    anonymus1

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