Plane and solid geometry . Given circle ABF of unit diameter, AB the side of a regularinscribed polygon of n sides, and CB the side of a regular in-scribed polygon of 2 n sides; denote AB by s and CB by x. To find X in terms of s. Argument 1. Draw diameter CF; draw BO and BF, 2. Z CBF is a rt. Z. 3. Also CF is the _L bisector of AB, i, .-. CB^ = CF CK. • 5. Now CF =1, BO == -, C0 = -. 2 2 6. •. cb^ = 0^ = 1 . CK = CK = CO — ko = ^ — KO. 2 . ^ 2 Reasons 1. §54,15. 2. §367. 3. §142. 4. §443,11. 5. By cons. 6. §309. 7. §447. 8. §54,13. 270 PLANE GEOMETRY Proposition XVI. Problem 567^ Given
Plane and solid geometry . Given circle ABF of unit diameter, AB the side of a regularinscribed polygon of n sides, and CB the side of a regular in-scribed polygon of 2 n sides; denote AB by s and CB by x. To find X in terms of s. Argument 1. Draw diameter CF; draw BO and BF, 2. Z CBF is a rt. Z. 3. Also CF is the _L bisector of AB, i, .-. CB^ = CF CK. • 5. Now CF =1, BO == -, C0 = -. 2 2 6. •. cb^ = 0^ = 1 . CK = CK = CO — ko = ^ — KO. 2 . ^ 2 Reasons 1. §54,15. 2. §367. 3. §142. 4. §443,11. 5. By cons. 6. §309. 7. §447. 8. §54,13. 270 PLANE GEOMETRY Proposition XVI. Problem 567^ Given a circle of unit diameter and the side of aregular circumscribed polygon of n sides, to find the sideof a regular circumscribed polygon of 2n Given circle 0 of unit diameter, AB half the side of a reg-ular circumscribed polygon of n sides, and CB half the sideof a regular circumscribed polygon of 2 n sides; denote AB by I and CB by |. To find X in terms of s. Argument 1. Dra\r CO and AO. ^ 2. /. BOC=\Z. BOA. 3. .-. in A OAB, AC : CB = AO : BO, 4. But AC = AB — CB, 5. And A o-v 6. .-. AB — CB : CB AB + BO- = ^AB^-{-~b7/: bo. 7. Substituting - for AB, - for CB, and - 2 2 2 AV2y 8. « — X : ic = V?^l : 1. Reasons 1. § 54, 15. 2. § 517, h, 3. §432. 4. § 54, 11. 5. §446. 6. §309. 7. §309. 8. §403. BOOK V Argument F. Reasons 9. 10. ., s — X = X Vs- -f 1. s 9. § Solving: 1 + Vs^ + 1 271 Ex. 1017. Given a circle of unit diameter and an inscribed and acircumscribed square ; compute the side of the regular inscribed and the regular circumscribed octagon. Proposition XVII. Problem 568. To compute the approximate value of the circum-ference of a circle in terms of its diameter; to computethe value of i
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