Sorry, don't know what you mean.
I'm looking for the coordinates of the blue point on the circle. I know another point (red dot) on the circle (2,3), the radius, and the arclength between the two points. I do not know the position of the centre of the circle
The co-ordinates of your blue point cannot be determined with the data given. There are an infinite number of different circles that have radius 10 and contain the point (2, 3). Any point on the circle using (2, 3) as the center with a radius of 10 is the potential center of a circle with radius 10 containing (2, 3). You have drawn just one of those possible circles, but have not told us how you determined it to be the correct circle.
Even if you knew the co-ordinates of the center, you would get two possible blue points, not a unique one.
Step1 would be to calculate central angle:
C = circle center, R = Red point, B = Blue point
u = central angleBCR (?)
a = arc length (given)
r = radius (given)
p = pi
u = 180a / (pr)
If r = 10 and arc = 9, you'll get u = ~51.57 degrees
Do you follow that?
Note:
sorry Skeeter; didn't see your post;
to OP: I'll leave the rest with Skeeter: he's beyond me...