Question: Can an ellipsoid be internally tangent to any tetrahedron?

This worksheet displays an ellipsoid internally tangent to the four sides of a tetrahedron.

The tetrahedron is a special case: it has a horizontal base with vertices A,B and C and its fourth vertex E is on the z axis.

However I have failed when trying to display an internally tangent ellipsoid in any other tetrahedron.

Will any tetrahedron support one or more internally tangent ellipsoids?

If so, are there conditions restricting the location of the mutual points of tangency?

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