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 4*PG* = *PQ*, where *G* is the centroid of the tetrahedron.

**Notes**:

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 3*PA'*.

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.