REGULAR POLYGONS. THE CIRCLE. MAXIMA AND MINIMA. REGULAR POLYGONS. 399. DEF. A regular polygon is a polygon which is both equilateral and equiangular; as, for example, the equilateral triangle and the square. Proposition 1. Theorem. 400. If the circumference of a circle be divided into any number of equal arcs, (1) the chords of these arcs form a regular polygon inscribed in the circle, and (2) the tangents at the points of division form a regular polygon circumscribed about the circle. Hyp. Let the Oce be divided into equal arcs at the pts. A, B, C, etc.; H let AB, BC, CD, etc., be chords of these ares, and GBH, HCK, etc., be с tangents. (1) To prove that ABCD .... is a K regular polygon. G B F E Proof. Since arc AB = arc BC = arc CD= etc., (Hyp.) ... chd. AB = chd. BC = chd. CD= etc., in the same o equal arcs are subtended by equal chords (194); and FAB = _ ABC = BCD= etc., all Z8 inscribed in equal segments are equal (239). ... ABCDEF is a regular polygon. (399) (2) To prove that GHK ... is a regular polygon. Proof. In the As ABG, BCH, etc., AB = BC = CD = etc., and _GAB _GAB = _GBA = _HBC = _HCB = etc., being measured by halves of equal arcs (243). ...the As ABG, BCH, etc., are all equal and isosceles. 401. COR. 1. If a regular inscribed polygon is given, the tangents at the vertices of the given polygon form a regular circumscribed polygon of the same number of sides. 402. COR, 2. If a regular in Α' Μ scribed polygon ABCD.... is given, the tangents at the middle points M, IN N, P, etc., of the arcs AB, BC, CD, etc., form a regular circumscribed polygon whose sides are parallel to those of the inscribed polygon, and whose vertices A', B', C', etc., lie on the radii OAA', OBB', etc. For, the sides AB, A'B' are 11, being I to OM, (204) and (210), and the same for the others; also, since B'M = B'N (268), the pt. B' must lie on the bisector OB (160) of the ZMON. 403. COR. 3. If the chords AM, MB, BN, etc., be drawn, the chords form a regular inscribed polygon of double the number of sides of ABCD.... 404. COR. 4. If through the points A, B, C, etc., tangents are drawn intersecting the tangents A'B', B'C', etc., a regular circumscribed polygon is formed of double the number of sides of A'B'C'D'.... 405. Sch. It is clear that the area of any inscribed polygon is less than the area of the inscribed polygon of double the number of sides ; and the area of a circumscribed polygon is greater than that of the circumscribed polygon of double the number of sides. Proposition 2. Theorem. 406. A circle may be circumscribed about a regular polygon, and a circle may be inscribed in it. Hyp. Let ABCDEF be a regular polygon. (1) To prove that a o may be circumscribed about it. Proof. Bisect the Zs A and B, and let the bisectors meet at 0. In the As OAB and OBC, AB = BC (399), OB is common, and ZOBA = ZOBC. ... A OBA = A OBC. (Cons.) (104) :: ZOAB = LOCB = } BCD, OA = OC. and In the same way it may be shown that each _ of the polygon is bisected by the line joining it with the pt. O, and that OA = OB = OC = OD = etc. .:. the o described with centre 0 and radius OA will pass through each of the angular points. (2) To prove that a o may be inscribed in ABCDEF. Proof. Since the sides AB, BC, etc., are equal chords of the circumscribed circle, they are equally distant from the centre. (206) Therefore, if a circle be described with the centre 0 and the perpendicular OH as a radius, this circle will be inscribed in the polygon: Q.E.D. 407. DEFS. The point 0, which is the common centre of the inscribed and circumscribed circles, is called the centre of the regular polygon. The radius of a regular polygon is the radius OA of the circumscribed circle. The apothem of a regular polygon is the radius OH of the inscribed circle. The angle at the centre of a regular polygon is the angle between two radii drawn to the extremities of any side, as AOB. 408. COR. 1. The inscribed and circumscribed circles of a regular polygon are concentric. 409. CoR. 2. The perpendicular bisectors of the sides of a regular polygon all pass through its centre. 410. COR. 3. The radius drawn to any vertex of a regular polygon bisects the angle at the vertex, 411. Cor. 4. If lines be drawn from the centre of a regular polygon to each of its vertices, the polygon will be divided into as many equal isosceles triangles as it has sides. 412. Cor. 5. Each angle at the centre of a regular polygon is equal to four right angles divided by the number of sides of the polygon. 413. Cor. 6. The interior angle of a regular polygon is the supplement of the angle at the centre. с Proposition 3. Theorem. 414. Regular polygons of the same number of sides are similar. Hyp. Let P and P' be two A B B' regular polygons of the same number of sides. F To prove that P and P' are similar. E' Proof. Since the polygons are regular, ... AB = BC CD = etc., and A'B' = B'C' = C'D' = etc. (399) AB BC CD A'B' B'C' C'D' E = etc. Also, if the polygons have each n sides, the sum of all the int. Zs of each polygon is (2n — 4) rt. Zs. (148) Since the polygons are equiangular (399), and have each the same number of sides, (Hyp.) 2п 4 ... each of each polygon rt. Zs. (150) n .:: LA = LA', B = _B', ZC= _0', etc. Hence the polygons are mutually equiangular and have their homologous* sides proportional. .. the polygons are similar. (307) Q.E.D. 415. COR. 1. The perimeters of regular polygons of the same number of sides are to each other as any two homolo (322) 416. COR. 2. The areas of regular polygons of the same number of sides are to each other as the squares of any two homologous* sides. (379) * Since the polygons are regular, any side of one may be taken as homologous to any side of the other. gous* sides. |