BOOK V. REGULAR POLYGONS. MEASUREMENT OF THE CIRCLE. 341. DEF. A regular polygon is a polygon which is both equilateral and equiangular. PROPOSITION I. THEOREM. 342. A circle can be circumscribed about, or inscribed in, any regular polygon. Let ABCDE be a regular polygon. I. To prove that a circle can be circumscribed about ABCDE. Let a circumference be described through the vertices A, B, and C (§ 223). Let O be the centre of the circumference, and draw OA, OB, OC, and OD. Then since ABCDE is equiangular, And since the triangle OBC is isosceles, ZOBC: LOCB. 360 16 ($ 91.) Then the circumference passing through A, B, and C also passes through D. In like manner, it may be proved that the circumference passing through B, C, and D also passes through E. Hence, a circle can be circumscribed about ABCDE. II. To prove that a circle can be inscribed in ABCDE. Since AB, BC, CD, etc., are equal chords of the circumscribed circle, they are equally distant from 0. (§ 164.) Hence, a circle described with O as a centre, and with the perpendicular OF from 0 to any side AB as a radius, will be inscribed in ABCDE. 343. DEF. The centre of a regular polygon is the common centre of the circumscribed and inscribed circles. The angle at the centre is the angle between the radii drawn to the extremities of any side; as AOB. The radius is the radius of the circumscribed circle; as OA. The apothem is the radius of the inscribed circle; as OF. 344. COR. From the equal triangles OAB, OBC, etc., we have ZAOB = Z BOC = 2 COD, etc. (§ 66.) Then each of these angles is equal to four right angles divided by the number of sides of the polygon. ($ 37.) That is, the angle at the centre of a regular polygon is equal to four right angles, divided by the number of sides. PROPOSITION II. THEOREM. 345. If the circumference of a circle be divided into any number of equal arcs, I. Their chords form a regular inscribed polygon. II. Tangents at the points of division form a regular circumscribed polygon. Let the circumference ACD be divided into any number of equal arcs, AB, BC, CD, etc. I. To prove ABCDE a regular polygon. = Now, chord AB = chord CD, etc. (§ 158.) chord BC arc BC: =arc CD, etc., we have arc BCDE =arc CDEA = arc DEAB, etc. Whence, LEAB = LABC = Z BCD, etc. Therefore, the polygon ABCDE is regular. (§ 193.) (§ 341.) II. Let FGHKL be a polygon whose sides LF, FG, etc., are tangent to the circle at the points A, B, etc., respectively. To prove FGHKL a regular polygon. In the triangles ABF, BCG, CDH, etc., we have Also, since are AB arc BC = arc CD, etc., we have LBAFL ABF = Z CBG = Z BCG, etc. (§ 197.) Hence, the triangles ABF, BCG, etc., are all equal and isosceles. ($$ 68, 94.) PROPOSITION III. THEOREM. 346. Tangents to a circle at the middle points of the arcs subtended by the sides of a regular inscribed polygon, form a regular circumscribed polygon. Let ABCDE be a regular polygon inscribed in the circle AC. Let A'B'C'D'E' be a polygon whose sides A'B', B'C', etc., are tangent to the circle at the middle points F, G, etc., of the arcs AB, BC, etc. To prove A'B'C'D'E' a regular polygon. We have, arc AB =arc BC =arc CD, etc. (§ 157.) Whence, arc AF =arc BF =arc BG = arc CG, etc. Therefore, arc FG =arc GH = arc HK, etc. Whence, the polygon A'B'C'D'E' is regular. (§ 345, II.) 347. COR. Let O be the centre of the circle, and draw OF, OL, and OA'. Then, OA' bisects / FOL. Whence, OA' passes through A. (§ 175.) (§ 154.) That is, the radii of a regular circumscribed polygon intersect the circumference in points which are the vertices of a regular inscribed polygon having the same number of sides. PROPOSITION IV. THEOREM. 348. Regular polygons of the same number of sides are similar. D Let A-E and A-E' be two regular polygons of the same number of sides. To prove A-E and A'-E' similar. The sum of all the angles of A-E is equal to the sum of all the angles of A'-E'. (§ 126.) Whence, each angle of A-E equals each angle of A'-E'. Again, since AB BC, etc., and A'B' = B'C', etc., A'B' B'C' C'D' Therefore, A-E and A'-E' are similar. PROPOSITION V. THEOREM. 349. The perimeters of two regular polygons of the same number of sides are to each other as their radii, or as their apothems. Let A-E and A'-E' be two regular polygons of the same number of sides. |