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Two points are randomly chosen inside a circle. Is it always possible to draw a straight line through each point, such that they subdivide the circle into 3 regions of equal area?

Bonus: can the lines subdivide the circle into 4 regions of equal area?

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1 Answer 1

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Unless I'm missing something obvious, the answer to the original question is:

No

To have only three regions, the two lines cannot intersect inside the circle, so the each line must independently divide the area into 1/3 and 2/3 regions.
But if one point is at the centre of the circle, any line through it bisects the circle, so cannot produce the required 1/3 region.

And the answer to the bonus question is:

Also No

Any adjacent pair of regions must sum to 1/2 the total area, and are separated by one of the two lines. From this it follows that both lines must bisect the circle.
All lines that bisect a circle pass through its centre, so both lines must do so. And to divide the circle into four equal regions, they must be perpendicular.
This will not be possible for most pairs of points, but for a specific counter-example, consider any pair of points on the circumference that are not 90 degrees apart.

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  • $\begingroup$ This is correct. Ok it was a little too easy. I've now added a bonus question. $\endgroup$ Jan 27 at 14:04
  • $\begingroup$ Sorry, missed your bonus answer which is quicker than mine. $\endgroup$
    – hexomino
    Jan 27 at 14:29
  • $\begingroup$ Why is this not a counterexample to your answer to the original question? i.stack.imgur.com/sJcTH.png Is there a requirement for what a "region" means that wasn't stated in the question? $\endgroup$ Jan 28 at 6:47
  • $\begingroup$ @JosephSible-ReinstateMonica Region wasn't defined as anything special/clever here, so I used the common/simple interpretation. I would not consider your two green areas (connected only at a corner) to be a single region, unless the OP explicitly defined region to allow that sort of solution. $\endgroup$
    – fljx
    Jan 28 at 9:54

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