This dynamic 3D applet does not work on Internet Explorer, and works best on a relatively new version of the free browser Firefox - click link to download latest version. It also requires the downloading & installation of the free Cabri 3D Plug In, available at Windows (4 Mb) or Mac OS (13.4 Mb).
Basic manipulation: 1) Right click (or Ctrl + click) and drag to rotate the whole figure (glassball).
2) Click to select and hold down the left button to drag P - the constant ratio PQ/PG is shown under Result. The shape of the tetrahedron can be changed by dragging any of the vertices A, B, C or D.
Or click Summary of manipulation to open & resize a separate window with instructions.
Sylvester's theorem for a tetrahedron: The resultant of four forces PA, PB, PC and PD acting on any point P with reference to a tetrahedron ABCD, is the force represented by 4PG, where G is the centroid of the tetrahedron.
1) One can determine the resultants of vectors with Cabri 3D. The resultant of PB and PC is shown by the pink vector, and the resultant of the pink vector with PD is shown by the yellow vector. Finally, the resultant of the yellow vector with PA is shown by the green vector PQ.
2) Also note that if we just consider triangle BCD and P as a point outside BCD in space, then the (yellow) resultant of vectors PB, PC and PD passes through the centroid of BCD, namely, A', and is equivalent to 3PA'.
See my paper Generalizing a problem of Sylvester for more details.
HTML export with Cabri 3D. Download a 30 day Demo, or for more information about purchasing this software, go to Cabri 3D.
Created by Michael de Villiers, 15 September 2012.