Dividing circle with (open ) curve

[gentleclockdivider]

Dividing a circle with an open curve gives a weird result , it performed the operation as the curve was closed 

 

[R C-R]

You get the weird result because annoyingly Affinity booleans ignore the stroke, only consider the area the curve covers, & closes open curves with straight line segments.

To do what you want, you need to give the open curve a stroke width (even a very tiny fraction of a pixel one) then use the Expand Stroke function to give it an already closed area, & then apply the divide.

[PixelPest]

Or make a shape from that curve: