Here is an outline of what needs to be done.

Lets start with what we know

Using this we can make a generic equation of of this plane note that

and

This gives the equation of the plane as

Note also since must be in the plane we get

(we will need this constraint later)

Note the volume of the tetrahedron is given by

Now we need to minimize this using Lagrange multipliers. This gives the system

Solving I get

This gives a volume of .