The two circles
O and O^\prime with radii of 1 are tangent with each other and with the horizontal line at the bottom. Circle O_1 is inscribed with a maximum possible radius inside O, O^\prime, and the horizontal line. O_2, O_3,... are successively inscribed following the same rule maximizing their radius.Find the diameter of a circle
O_n.Previous iteration: #720410 (quite a few different valid solutions were found for this one, congrats~~)
O_1looking at the right triangle(1-r)^2 + 1^2 = (1+r)^2O_1is\frac{1}{2}.O_2giving\frac{1}{6}andO_3giving\frac{1}{12}etc.O_nO_3though.O_2(1 - \frac{1}{2} - r)^2 + 1 = 1 + 2*r + r^2O_3(1 - \frac{1}{2} - \frac{1}{6} - r)^2 + 1 = 1 + 2*r + r^2(\frac{6}{6} - \frac{3}{6} - \frac{1}{6} - r)^2 + 1 = 1 + 2*r + r^2(\frac{1}{3} - r)^2 + 1 = 1 + 2*r + r^2\frac{1}{9} - \frac{2}{3}*r + r^2 + 1 = 1 + 2*r + r^2\frac{1}{9} - \frac{2}{3}*r = 2*r\frac{1}{9} = \frac{8}{3}*r\frac{3}{9*8} = r\frac{1}{24} = r\frac{1}{12}\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, ...rto denote the different radii ofO_1(as in the drawing) andO_2(as in the right-hand side of the formula below)? And same comment for the subsequentO_3equation?1/18forO_3.O_3O)r, whererdenotes the radius ofO_3.O_n).1minus the sum of all diametersO_1andO_2minusrofO_3.rofO_3.