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<\/figure><\/div>\n\n\n\nTo solve the problem, we can draw various lines as the following so that we can have a clear picture:<\/p>\n\n\n\n
![\"\"](\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-23-at-3.13.26-PM.png\")
<\/figure><\/div>\n\n\n\nAssume the radius of the circles is $r$. Obviously, $\\triangle{EFG}$ is equilateral as \n\n$$\\overline{EF}=\\overline{FG}=\\overline{GE}=2\\cdot r$$\nTherefore, \n$$\\overline{EK}=\\dfrac{\\sqrt{3}}{2}\\cdot\\overline{FG}=r\\cdot\\sqrt{3}$$\nSince \n$$ \\overline{AE} + \\overline{EK} + \\overline{KN} + \\overline{NC} = \\sqrt{2}$$\nwe have \n$$r\\cdot\\sqrt{2} + r\\cdot\\sqrt{3} + r + r\\cdot\\sqrt{2}=\\sqrt{2}$$\nTherefore\n$$r=\\frac{\\sqrt{2}}{1 + 2\\sqrt{2}+\\sqrt{3}}=\\frac{2}{4+\\sqrt{2}+\\sqrt{6}}$$ $$ = \\frac{1}{2}(8-5\\sqrt{2}+4\\sqrt{3}-3\\sqrt{6}) \\approx 0.2543331$$<\/p>\n\n\n\n
Additional information about circle packing in a square can be found at here<\/a> or here<\/a>.\n Today, we talk about a classic problem of packing circles in a square. Given a unit square, you need to make 3 congruent, non-overlapping circles that are as big as possible. It has been proven that the arrangement of 3 circles must be as the following. Can … Continue reading