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 any of the four vertices of the tetrahedron.
Or click Summary of manipulation to open & resize a separate window with instructions.
The face angle bisector in space is the analogue of an angle bisector in the plane. Hence, a face angle bisector is the set of points equidistant from two faces, or equivalently, it is the plane that bisects the angle between two faces.
Theorem 2: The six face angle bisectors of a tetrahedron are concurrent (at the incentre of the tetrahedron, and therefore the tetrahedron has an insphere)¹.
Challenge: Can you explain (prove) why this theorem is true? (Hint: use the idea of equi-distance.) Only if stuck, go to 3D Concurrency Proofs.
¹Note that in this applet the incentre may sometimes fall outside, and will then produce an ex-sphere. But just like the triangle has an ex-centre and ex-circle associated with each side, the tetrahedron also has an ex-centre and ex-sphere associated with each of its four faces.
HTML export by Cabri 3D. Download a 30 day Demo, or for more information about purchasing this software, go to Cabri 3D.
Created by Michael de Villiers, 26 April 2012; updated 21 April 2024.