{"problem":{"name":"Ex - Disk and Segments","description":{"content":"There are $N$ line segments in a coordinate plane, and the $i$\\-th line segment $(1\\leq i\\leq N)$ has two points $(a _ i,b _ i)$ and $(c _ i,d _ i)$ as its endpoints. Here, each line segment includes ","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc314_h"},"statements":[{"statement_type":"Markdown","content":"There are $N$ line segments in a coordinate plane, and the $i$\\-th line segment $(1\\leq i\\leq N)$ has two points $(a _ i,b _ i)$ and $(c _ i,d _ i)$ as its endpoints. Here, each line segment includes its endpoints. Additionally, no two line segments share a point.\nWe want to place a single closed disk in this plane so that it shares a point with each line segment. In other words, we want to draw a single circle so that each line segment shares a point with either the circumference of the circle or its interior (or both). Find the smallest possible radius of such a disk.\n\n## Constraints\n\n*   $2\\leq N\\leq 100$\n*   $0\\leq a _ i,b _ i,c _ i,d _ i\\leq1000\\ (1\\leq i\\leq N)$\n*   $(a _ i,b _ i)\\neq(c _ i,d _ i)\\ (1\\leq i\\leq N)$\n*   The $i$\\-th and $j$\\-th line segments do not share a point $(1\\leq i\\lt j\\leq N)$.\n*   All input values are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$\n$a _ 1$ $b _ 1$ $c _ 1$ $d _ 1$\n$a _ 2$ $b _ 2$ $c _ 2$ $d _ 2$\n$\\vdots$\n$a _ N$ $b _ N$ $c _ N$ $d _ N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc314_h","tags":[],"sample_group":[["4\n2 3 2 10\n4 0 12 6\n4 8 6 3\n7 8 10 8","3.319048676309097923796460081961\n\nThe given line segments are shown in the figure below. The closed disk shown in the figure, centered at $\\left(\\dfrac{32-\\sqrt{115}}4,\\dfrac{21-\\sqrt{115}}2\\right)$ with a radius of $\\dfrac{24-\\sqrt{115}}4$, shares a point with all the line segments.\n![image](https://img.atcoder.jp/abc314/cbcd8322e610eefca04d6f5a7ddbc89a.png)\nIt is impossible to place a disk with a radius less than $\\dfrac{24-\\sqrt{115}}4$ so that it shares a point with all the line segments, so the answer is $\\dfrac{24-\\sqrt{115}}4$.\nYour output will be considered correct if the absolute or relative error from the true value is at most $10^{-5}$, so outputs such as `3.31908` and `3.31902` would also be considered correct."],["20\n0 18 4 28\n2 21 8 21\n3 4 10 5\n3 14 10 13\n5 9 10 12\n6 9 10 6\n6 28 10 18\n12 11 15 13\n12 17 12 27\n13 17 20 18\n13 27 19 26\n16 1 16 13\n16 22 19 25\n17 22 20 19\n18 4 23 4\n18 5 23 11\n22 16 22 23\n23 15 30 15\n23 24 30 24\n24 0 24 11","12.875165712523887403637822024952\n\nThe closed disk shown in the figure, centered at $\\left(\\dfrac{19817-8\\sqrt{5991922}}{18},\\dfrac{-2305+\\sqrt{5991922}}9\\right)$ with a radius of $\\dfrac{3757\\sqrt{29}-44\\sqrt{206618}}{18}$, shares a point with all the line segments.\n![image](https://img.atcoder.jp/abc314/6f259a531d06b430c5dc1299c4d2ecdd.png)"],["30\n526 655 528 593\n628 328 957 211\n480 758 680 794\n940 822 657 949\n127 23 250 385\n281 406 319 305\n277 598 190 439\n437 450 725 254\n970 478 369 466\n421 225 348 141\n872 64 600 9\n634 460 759 337\n878 514 447 534\n142 237 191 269\n983 34 554 284\n694 160 589 239\n391 631 22 743\n377 656 500 606\n390 576 184 312\n556 707 457 699\n796 870 186 773\n12 803 505 586\n343 541 42 165\n478 340 176 2\n39 618 6 651\n753 883 47 833\n551 593 873 672\n983 729 338 747\n721 77 541 255\n0 32 98 597","485.264732620930836460637042310401"]],"created_at":"2026-03-03 11:01:14"}}