## Monday, February 28, 2011

### Fox 331

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

1. E midpoint of AB;F midpoint of CD
G midpoint of EF

2. No, this won't work.

3. 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

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

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...

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.

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

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).

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

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

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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