四面體與圓球體

The answer is yes.

First we rotate the tetrahedral ABCD that one of the plane ABC is parallel to the xy-plane in cartesian coordinate system.

Draw a line L that passing through the circumcentre, perpendicular to the xy-plane. Then all points on L is equaldistant to ABC.

Now we shall prove that there exist a point P on L that PA=PB=PC=PD.
On one of the side and tends to infinitely far, PD > PA=PB=PC.
And on another side tends to infinitely far, PD < PA=PB=PC.
Since change in length is differentiable at any point on L, by fixed point theorem/mean value theorem, there exist a point P that PD = PA = PB = PC.
And since the distance (PD-PA) behaves monotonically, we can conclue that the sphere that touches all 4 vertices of a tetrahedral is unique.