See Regular Tetrahedron 1234.

Construct an equilateral triangle such that its vertices and circumcentre are (in any order) 1,2,3, and 4 units from the origin.

What is the length of a side of the triangle?

Dej,

Would it be possible for you to give coordinates for the vertices of the equilateral so drawn, so that I can graph it for myself? The triangle I attempted to draw based on your answer did not have sides of equal length.