Intersection of chords when a point is circling

BACKGROUND

The image below illustrates 4 chords in a circle with origo O.
N is a point on that circle.
So:
angle KNM is constant for all N (wherever N is located on the circle).
angle KNL = MNL for all N.
(and N is here always "above" KM, i.e. on the larger of the arcs)

Attachment 26760

The image below introduces N2 on the same circle as N1 (N above).
It also introduces the intersection point P on the chord KM.
Illustrated are:
P1 where the lines LN1 and KM intersect.
P2 where the lines LN2 and KM intersect.
Attachment 26761

C is the middle point between K and M, i.e. CK=CM.
Everything remains contant except the circling point N.


QUESTION

How does the distance CP vary with the location of N on the circle?
(CP is the distance between point C and point P, to be clear)

Any kind of formulation of the relationship between where point N is, and the distance CP, would be very helpful.

What frustrates me here is that the angle KNM (i.e. with apex in N) is the same, no matter where it is on this circle. So it really holds no information at all as to where point N actually is located. (I would like to call this "Euclid's curse"!) But I think that its relationship to CP might be the key. CP must be unique for every N (known to be on the larger arc). So if I know CP then I will find N and vice versa.

Oh, quite some views on this post, but no answers. Is it that difficult? Yeah well it beats me too.
I'll have to run it numerically then. Analytical solutions are for kiddies and their school books...
:-D
(But this did look so simple)

Let the radius of the circle be r and the length of OC be c, then CL = r-c.
The angle COM will be given by cos(COM)=c/r.
Define the position of the point N by means of the angle that ON makes with OM. Call it
You can now calculate the angle CPL in terms of those two angles and its tangent will be equal to (r-c)/CP.
That gets you CP in terms of
You will still need to investigate what happens when and after ON aligns with OL.

Thank you!
But I would need clarification on how CPL can be determined by theta (and r and c).
It is tangent of (r-c)/CP, but I don't know CP, nor theta.

Your idea with the triangle NOM seems ingenius. But I don't get from there to triangle CPL. I'm just a bit too daft to follow the line all the way here...