From: Doctor Nisith Bairagi firstname.lastname@example.org Uttarpara, West Bengal, India.
Subject Reference: ?To divide an angle into any number of equal parts? suggested by: Shyamal Kumar Das <email@example.com> in Math Forum.
Congratulations Shyamal! Your method of dividing an angle into any number of equal parts, works. You have only shown to get one-fifth of 180 deg, and remarked that the very method is valid for any angle into any number of parts, without providing proof of its generality. Following the steps of your method, I supplement the Proof for its generality. Refer to your Figure of division of 180 deg into 5 equal parts.
Method Recollected: Let angle X deg is to be divided equally into ?n? number of parts, to get angle x = X/n. Depending on the accuracy needed, the angle X is to be initially divided equally into large number of parts ?k?, (say 8, 16, 32 , ?etc), to get the line BN corresponding to X/k line. BD = r, BH = n.r, The arc DM makes angle ?a? at B, so that angle MBD = angle NBH = a, and arc HI = arc DM = r.a. On the bigger arc (radius of which is BH = n.r), arc HR = k.MD = k.HI. Thus, angle RBH = x = X/n.
Proof of Generalisation: a = X/ k, arc HI = arc DM = r.a arc HR = k.arc HI = k.r.a = k.r.(X/k) = r.X So, angle RBH = arc HR / radius BH = (r.X)/(n.r) = X/n. Hence, x = X/n (Proved) [Das took X = 180 deg, n = 5, k = 8, so that x = X/n = 180/5 = 36 deg.]
Remarks : The method as suggested by Shyamal Kumar Das though is approximate, but it works well for any angle X to be divided equally into any number of parts n. The accuracy of the method increases with higher values of k, like, 8, 16, 32,? and their multiples, as suggested. To Dr Math : Please display/post this Generalised form in your column, and in the Discussion column for Shyamal Kumar Das, and readers, and acknowledge with comments to my Email address.