I have a question:
There is a tetrahedron with coordinates $(3,1,5),(-7,0,13),(27,5,-13),(-17,4,21)$ that is inscribed in one sphere. What is the minimum distance needed to "walk" to each point once without a diagonal intersection (to the nearest hundredth since I don't want messy answers)?
I created this question to test my geometry.
I figured I could use the Law of Cosines with Extended Law of Sines to find the circumcircle of any set of three points, but I can't figure out where to go from those circles.
Right now, I need to find where the sphere is, and how to find the minimum perimeter from there.
Note: There is actually one. Oops.
Update: As Narlin has shown, the sphere's center is $(393,\frac{69}2,2507),$ with an radius of approximately $636.57383\dots.$ The task is now down to finding the minimum perimeter.
I translating the sphere and tetrahedron such that the origin is the center of the sphere is useful. Now it's just a matter of finding the shortest path.
All we care about is the angle measurements between points (and also intersections).
I can use the vector formula $\textbf{v}\bullet\textbf{w}=||\textbf{w}||||\textbf{v}||\cos{\theta},$ but this, along with checking combinations, seems a bit too long to me. Is there a way to use a shortest distance result, or is my method the best?