Split curves at apparent intersection based on current view

[vano_artful (V@no)]

Hello.
I have 3 curves and rotated the view, so all 3 curves appear like intersecting (overlapping) each other at the same spot, then set cplane to view:

When I attempt to split curves at the point they visibly overlap, rhino says they are not within tolerance of each other.
Ok, so I added a point on one of the curves at the apparent intersection and when I tried use that point to cut other curves with, it was showing cut placement at the wrong place:

Why is that? What am I doing wrong?

Thank you

[vano_artful (V@no)]

My current solution is to ExtrudeCrv the curves and use created surface to cut other curves with.
It works, but I feel like there should be an easier way to do this.