The sphere will be the one for which each of the planes of the tetrahedron are tangential (to it), i.e. the one for which the perpendicular distances from the sphere centre to each of the planes of the tetrahedron are equal.

Flipping the tetrahedron upside down, so as to avoid the irritating negative z co-ordinates, the equation of the ABC (new C) plane will be x+y+z/2=1, and the perpendicular distance from an arbitrary point (X,Y,Z) on the origin side onto this plane will be (1-X-Y-Z/2)/sqrt(1+1+1/4) = (2/3)(1-X-Y-Z/2). The distances from (X,Y,Z) onto the other three planes will be X, Y and Z. Equating the four leads to a radius of 1/4.