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Unit circles through three of n points
Show that n points in the plane have only o(n²) unit circles passing through three of them; the best construction gives n^{3/2}.
| 1 | { |
| 2 | "domain": "mathematics", |
| 3 | "topics": [ |
| 4 | "discrete-geometry", |
| 5 | "combinatorics" |
| 6 | ], |
| 7 | "type": "proof", |
| 8 | "also": [ |
| 9 | "construction" |
| 10 | ], |
| 11 | "cover": { |
| 12 | "motif": "points", |
| 13 | "n": 8 |
| 14 | }, |
| 15 | "source": { |
| 16 | "key": "erdos", |
| 17 | "name": "Erdős Problems", |
| 18 | "url": "https://www.erdosproblems.com/104", |
| 19 | "id": "104", |
| 20 | "curator": "Thomas Bloom", |
| 21 | "poser": "Paul Erdős", |
| 22 | "status": "open, $100 prize", |
| 23 | "as_of": "2026-10-20", |
| 24 | "prize": "$100", |
| 25 | "catalog": "0521c19bbd38" |
| 26 | }, |
| 27 | "review": true, |
| 28 | "approval": "first-time" |
| 29 | } |