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I'm given one of the vertices of a regular tetrahedron and the radius of the circumsphere. I also know the center point of the circumsphere. How can I find the remaining three vertices? (It was pointed out in the comments that "given any solution, you can rotate it around the line through the given vertex and the circumcentre." I think this doesn't matter since any possible solution will work for my application).

So far I've mostly just tried guessing. I'm working on an OpenGL program and I need to draw a regular tetrahedron algorithmically. I don't have much of a math background so I'm a bit lost on even where to begin on this.

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    Mostly just guessing. I'm working on an OpenGL program and I need to draw a tetrahedron algorithmically. I don't have much of a math background so I'm a bit lost on even where to begin on this.2012-01-07
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    This is not homework. It's for a programming project (this project is also not homework) I'm working on. I'm just under-mathed is all. :(2012-01-07
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    OK, thanks for the clarifications. It might be a good idea to edit the question and state these explicitly, for the benefit of others.2012-01-07
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    This is strongly underspecified. You can take any point whose distance from the given vertex is the given radius, and then take any three vertices (not coplanar with the given vertex) on the sphere around that point with that radius.2012-01-07
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    @joriki I've updated the question to reflect that I need an isosceles tetrahedron.2012-01-07
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    I need to be more thorough with this ... I also know the center point of the circumsphere.2012-01-07
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    Are you sure you mean an [isosceles tetrahedron](http://mathworld.wolfram.com/IsoscelesTetrahedron.html) and not a [regular one](http://en.wikipedia.org/wiki/Regular_polyhedron)? I was half suspecting that you might have these additional constraints, but even then it's underspecified -- given any solution, you can rotate it around the line through the given vertex and the circumcentre.2012-01-07
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    Hmmm.. I think maybe you are right that I mean a regular tetrahedron. However, I'm not sure I even entirely understand the different between a regular and an isosceles tetrahedron. But as far as the rotation, any solution works for my application.2012-01-07
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    I think regular tetrahedron is what I'm after ... updating the question again ...2012-01-07
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    Here is an ugly recipe. Let your given vertex be $V$, your centre be $C$, and let the circumsphere radius be $r$. $(1)$ Find a nice regular tetrahedron $T$ with center the origin and radius of circumsphere $r$; $(2)$ Translate $C$ to the origin; $(3)$ Find a rotation that takes one of the vertices of $T$ to the translated version of $V$. There are many; $(4)$ Compute what this rotation does to the remaining vertices of $T$; $(5)$ Translate back.2012-01-07

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