Dividing Line

Yes. Draw a line ℓ that doesn’t intersect the circle, and choose a point P on it that doesn’t lie on any of the lines determined by pairs of points within the circle. Rotate ℓ about P. It will pass through all the points in the circle one at a time (because it can’t encounter two at a time). Stop when it’s passed 50,000 points.

From Claudi Alsina, Charming Proofs, 2010.