I am trying to solve for the 4th vertex of a tetrahedron with 4 known vertices. The vertices of the tetrahedron are the centers of 4 spheres. Basically I need to find the coordinates of the 4th sphere whose radius is also already known. A set of formulas are actually given to solve this but I can't seem to figure out how to use them: $$(x_d-x_a)^2 + (y_d-y_a)^2 + (z_d-z_a)^2=(r_d+r_a)^2$$ $$(x_d-x_b)^2 + (y_d-y_b)^2 + (z_d-z_b)^2=(r_d+r_b)^2$$ $$(x_d-x_c)^2 + (y_d-y_c)^2 + (z_d-z_c)^2=(r_d+r_c)^2$$ a, b, and c represent the circles of known positions and the d is the one whose x,y,z coordinates need to be determined. Again, all radii of all 4 spheres are known.
The system would look like this: sphere bounded tetrahedron
Thank you very much in advance!