> I don't think that's true: Constraint 1. is sufficient.

True in general, But the way I define what D _IS_, is that it is the
circumcenter of ABC offest in the normal direction by some distance H.
That way the whole system is parameterized so that for any triangle ABC, I can
vary how much of sphere S "pokes through" circle O.


> Every non-degenerate triangle has exactly one circumscribed circle, so
> obviously there exists exactly one circle touching ABC. Let O be that
> circle.
>
> Let N be the rotational axis of O. If D is on N, Let P be an arbitrary
> plane trough N; otherwise, let P be the plane through D and N. Note that
> in either case both D and N are on P.
>
> Note that P intersects O in two points; let these be E and F. Also note
> that D is not on the line EF, otherwise ABCD would be coplanar. Thus,
> DEF is a non-degenerate triangle, and itself has a circumscribed circle.
> Let Q be that circle.
>
> Note that E and F have equal distance to N, thus N runs through the
> center of Q, so rotating Q about N defines a sphere. Let S be that
> sphere. Note that D lies on Q and hence on S; also note that E and F
> both lie on Q, and their rotation about N yields O, thus O lies on S.
> Note that ABC all lie on O, and therefore also on S.
>
> => ABCD all lie on S.
>
> Finally note that if P if ambiguous, the case is entirely rotationally
> symmetric except for the positions of ABC, and thus S is the same for
> all possible choices of P.
>
> => S is unambiguous.
>
> => For any non-degenerate tetrahedron ABCD, there exists exactly one
> circumscribed sphere.

I did manage to follow all of that after 3 or 4 tries, and you are of course,
correct.   Oh, if only my mind still worked as clearly as yours.

After a few false starts, I got it all to work out and am cleaning up all of the
accumulated dross in the code.

I must say I was rather surprised how little information is readily available
(internet-wise) on practically accomplish this otherwise-simple task.

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