tag:blogger.com,1999:blog-6500033298667240354.post347411484318825406..comments2024-02-19T00:34:12.578-08:00Comments on Always Creative Geometry Problems plus Occasionally Annoying Philosophy: Fox 3318foxeshttp://www.blogger.com/profile/09567328431908997738noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-6500033298667240354.post-6939789486738258192011-03-10T13:09:44.575-08:002011-03-10T13:09:44.575-08:00this simple construction of G3
M midpoint of diago...this simple construction of G3<br />M midpoint of diagonal BD<br />O intersection of the diagonals<br />on diagonal AC,AN=CO<br />then MG=1/3 MN<br />Essai sur la determination des centres de gravité<br />Hyacinthe Celestin Gaubert (1839)p27<br />anonymus1Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-62529984883881905212011-03-08T11:26:35.200-08:002011-03-08T11:26:35.200-08:00The name of the guy is Jim Wilson. Sorry about th...The name of the guy is Jim Wilson. Sorry about that...Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-43279539711414685042011-03-07T13:29:11.328-08:002011-03-07T13:29:11.328-08:00There is a nice animation in Joe Wilson's page...There is a nice animation in Joe Wilson's page.<br />See here:<br /><br />http://jwilson.coe.uga.edu/emt668/EMT668.Folders.F97/Patterson/EMT%20669/centroid%20of%20quad/Centroid.htmlAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-16736000823561148532011-03-04T07:03:22.514-08:002011-03-04T07:03:22.514-08:00The centroid of the vertices of a quadrilateral oc...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).<br /><br />http://mathworld.wolfram.com/Quadrilateral.htmlCésar Lozadanoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-57194529349150667352011-03-03T21:16:57.793-08:002011-03-03T21:16:57.793-08:00I2
the centroid of a massive line is the midpoint ...I2<br />the centroid of a massive line is the midpoint <br />with mass proportional to the length l<br />let be A',B',C',D' the midpoints of the sides a,b,c,d<br />we must find K center of the point masses (A',a),(B',b)<br />A'K=a/(a+b)A'B'<br />this can be done with parallel lines<br />L center of (C',c)and (D',d)<br />G2 is the centroid of (K,a+b),(L,c+d)<br /><br />I3<br />the center of gravity of an homogenious triangle is the centroid with a mass proportional to the aera<br />each diagonal divides ABCD into 2 triangles<br />with centroids C1,C2(first diagonal),C3,C4(second<br />diagonal)<br />G3 lies on the lines C1C2 and C3C4<br />G3 is the intersection of this lines<br />(thank you for the hints)<br />anonymous1Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-5139209884805914532011-03-03T11:23:37.136-08:002011-03-03T11:23:37.136-08:00"I highly suspect that initial 8foxes intenti..."I highly suspect that initial 8foxes intention was I3. Anyway, all three interpretations have a construction solution..."<br /><br />Correct. Slight coloring of the quadrilateral implies that.8foxeshttps://www.blogger.com/profile/09567328431908997738noreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-10071750682785143922011-03-03T11:12:04.823-08:002011-03-03T11:12:04.823-08:00hmmm ok! there are three possible interpretations ...hmmm ok! there are three possible interpretations of problem statement:<br />I1) centroid of 4 equal 0-dimensional masses at quadrilateral vertices<br />I2) centroid of 4 homogeneously massive 1-dimensional lines forming the given quadrilateral<br />I3) centroid of a homogeneously massive 2-dimensional quadrilateral.<br /><br />@Anonymous 1: your solution solves I1 only<br /><br />I highly suspect that initial 8foxes intention was I3. Anyway, all three interpretations have a construction solution...Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-40134288433268194582011-03-03T02:04:59.015-08:002011-03-03T02:04:59.015-08:00theorem
the centroid of the vertices of a quadrila...theorem<br />the centroid of the vertices of a quadrilateral<br />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<br />anonymous 1Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-40046332765246931922011-03-03T01:50:10.997-08:002011-03-03T01:50:10.997-08:00the first answer works
other solutions:
I midpoint...the first answer works<br />other solutions:<br />I midpoint of AD;J midpoint of BC<br />G midpoint of IJ<br />or<br />K midpoint of AC,L midpoint of BD<br />G midpoint of KL<br />and the 3 ways give the same point G!<br />anonymous1Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-32178823062290467642011-03-02T10:35:12.864-08:002011-03-02T10:35:12.864-08:00No, this won't work.No, this won't work.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-32882607148086980592011-03-02T07:29:21.585-08:002011-03-02T07:29:21.585-08:00ABCD the quadrilaterl
E midpoint of AB;F midpoint ...ABCD the quadrilaterl<br />E midpoint of AB;F midpoint of CD<br />G midpoint of EFAnonymousnoreply@blogger.com