I needed something similar some time ago. Actually I needed to check an open curve for Inside containment, but the principle is similar with Coincident one.
There might be a better way, but I succeeded by basically closing the open curve, and then checking for Containment.
That said, I found curve.MakeClosed method to be really tricky in working with curves that are not polylines (curve degree 1).
A workaround may be to recreate your open curve as an interpolated closed curve. Then check for point containment.
Sorry for not being able to help you with C#, this is python example instead:

This is a good idea, however my curves are lines. Are there some function to find out if on point is contained in a line? For example from the equation of the line.

I do not know, sorry, will leave that to more math knowledgeable guys around here.

If you decide to go with the rhinocommon methods, you can use line.DistanceTo(anotherCurve.PointAtStart, True) with limitToFiniteSegment input parameter set to True to check for the finite line segment only.