Dividing Line

Draw a circle and choose 100,000 points at random in its interior. Is it always possible to draw a line through the circle such that 50,000 points lie on each side of it?

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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.

December 10, 2025December 9, 2025 | Puzzles