The Modal Ontological Argument - Without Modal Logic

[Rational AKD]

Maybe you should re-read the definition of coplanar

In geometry, a set of points in space is coplanar if all the points lie in the same geometric plane. For example, three distinct points are always coplanar; but a fourth point or more added in space can exist in another plane, or, incoplanarly.
http://en.wikipedia.org/wiki/Coplanarity

yes, 3 points are always coplanar. however, in the example you gave the lines (though they aren't technically lines) do not remain in the same plane. they curve outside the plane of the three points. because the lines curve they are not line segments, and because they curve outside the plane of the 3 points they are not coplanar either.