For a regular tetrahedron with side length $s,$ show that the radius of the circumsphere of the tetrahedron is $\frac{\sqrt{6}}{4}s.$
For starters, I know that the centroid of a tetrahedron splits its medians into ratios of $3:1.$ I took a look at the following link: http://www.mathematische-basteleien.de/tetrahedron.htm but I didn't really understand how they got the medians were split by the centroid into ratios of $2:1.$ (I understand that they are considering a $2$ dimensional triangle to prove it but I don't understand how this works). Furthermore, how do we necessarily know the centroid of the tetrahedron is the center of its circumcircle and how do we know that the radius of the circumsphere is $\frac{2}{3}$ of the height? May I have some assistance to understand what they are doing? Thanks in advance.