I am stuck finding my ways around the questions. I have ideas about them but they are not coming forth! I will appreciate if anyone can help you.

my questions go thus:

I am working in 3 dimensions over real number R, and I am considering the equilateral triangle in xy-plane with vertices A(0,0,0), B(1,0,0) and C(1/2, 3 /2, 0). So I need to find

(i). the point D(x,y,z) in the first octant so that T(ABCD) is a regular tetrahedron.

(ii). the centroid P of T(ABCD).

(iii) the equation of sphere inscribed in T(ABCD) that is centered at P.

and finally, the equation of the ellipsoid inscribed in the unit tetrahedron with vertices (0,0,0), (1,0,0), (1,0,0) and (0,0,1) where the ellipse intersects the centroids of the faces.

I will gladly appreciate any suggestions or help on these problems. I have tried working on it but not getting desired solutions and ashamed of even sharing my solutions. I never believed I would struggle with it. Thank you in advance!!!