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)

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.

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 CPvary 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.