Outlining a tetrahedron (don't ask why!), I have sketched fully-defined triangles on the three principle planes, all coincident at the origin and each of the three pairs coincident at one other point that will become another vertex of the tetrahedron. (I seem to have no problem with coincidence of the corners of those triangles among the different planes, though now I am wondering why.) Then I defined an oblique plane determined by the three outer corners of these pairs (or at least by the end of two of the four lines coincident there). I sketched the triangle on that plane that is determined by these same three points to close the tetrahedron. This fourth triangle also says it's fully defined (see image below).

Now the specific questions:

1) When I blow up any one of the corners of this fourth triangle far enough, it does not appear coincident with the corners of the other two triangles that do appear coincident there (see image below).

Why is this?

2) In general, how do you apply, and how do you verify, relations among sketches on different planes, in this case three sketches on three planes? This would usually be an issue on the lines where the planes happen to intersect (or the points where three planes intersect, as in this case).

3) Finally, if the corners of the fourth triangle actually are coincident with the those of the other two (in each case), why do they not appear so?

-- John Willett

You're right on 'coincident' can pick up points not from the same plane by projecting them, that's one of it's uses. In your case, you used sketch points to define the plane, then made endpoints of a sketch line coincident to these points (that were used to define the plane). So in your case the endpoints of the sketch lines from different sketches do meet at the same point.