therefore, taking the letters P, Q and P', to denote the areas of the but, the polygons P and P' being similar, we have, by (23), Therefore, the polygon P' is similar to the polygon and equiva lent to the polygon Q, as required. REGULAR POLYGONS. MEASUREMENT OF THE CIRCLE. MAXIMA AND MINIMA OF PLANE FIGURES. REGULAR POLYGONS. 1. DEFINITION. A regular polygon is a polygon which is at once equilateral and equiangular. The equilateral triangle and the square are simple examples of regular polygons. The following theorem establishes the possibility of regular polygons of any number of sides. PROPOSITION I.-THEOREM. 2. If the circumference of a circle be divided into any number of equal parts, the chords joining the successive points of division form a regular polygon inscribed in the circle; and the tangents drawn at the points of division form a regular polygon circumscribed about the circle. Let the circumference be divided into the equal arcs AB, BC, CD, etc.; then, 1st, drawing the chords AB, BC, CD, etc., ABCD, etc., is a regular inscribed polygon. For, its sides are equal, being chords of equal arcs; and its angles are equal, being inscribed in equal segments. G B H K D 2d, Drawing tangents at A, B, C, etc., the polygon GHK, etc., is a regular circumscribed polygon. For, in the triangles AGB, BHC, CKD, etc., we have AB = BC CD, etc., and the angles GAB, GBA, HBC, HCB, etc., are equal, since each is formed by a tangent and chord and is measured by half of one of the equal parts of the circumference = (II. 62); therefore, these triangles are all isosceles and equal to each other. Hence, we have the angles G = H = K, etc., and AG = GB = BH = HC CK, etc., from which, by the addition of equals, it follows that GH HK, etc. = 3. Corollary I. Hence, if an inscribed polygon is given, a circumscribed polygon of the same number of sides can be formed by drawing tangents at the vertices of the given polygon. And if a circumscribed polygon is given, an inscribed polygon of the same number of sides can be formed by joining the points at which the sides of the given polygon touch the circle. It is often preferable, however, to obtain the circumscribed polygon from the inscribed, and reciprocally, by the following methods: E B' F C' B G D D 1st. Let ABCD.... be a given inscribed polygon. Bisect the arcs AB, BC, CD, etc., in the points E, F, G, etc., and draw tangents, A'B', B'C', C'D', etc., at these points; then, since the arcs EF, FG, etc., are equal, the polygon A'B'C'D' .... is, by the preceding propo'tion, a regular circumscribed polygon of the same number of sides as ABCD.......... Since the radius OE is perpendicular to A AB (II. 16) as well as to A'B', the sides A'B', AB, are parallel; and, for the same reason, all the sides of A'B'C'D'.... are parallel to the sides of ABCD.... respectively. Moreover, the radii OA OB, OC, etc., when produced, pass through the vertices A,B, C', etc.; for since B'E B'F, the point B' must lie on the line OB which bisects the angle EOF (I. 127). 2d. If the circumscribed polygon A'B'C'D'.... is given, we have only to draw OA', OB', OC', etc., intersecting the circumference in A, B, C, etc., and then to join AB, BC, CD, etc., to obtain the inscribed polygon of the same number of sides. 4. Corollary II. If the chords AE, EB, BF, FC, etc., be drawn, a regular inscribed polygon will be formed of double the number of sides of ABCD.... If tangents are drawn at A, B, C, etc., intersecting the tangents A'B', B'C', C'D', etc., a regular circumscribed polygon will be formed of double the number of sides of A'B'C'D'.... It is evident that the area of an inscribed polygon is less than that 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 II.-THEOREM. 5. A circle may be circumscribed about any regular polygon; and a also be inscribed in it. circle may B H E D Let ABCD... be a regular polygon; then, 1st. A circle may be circumscribed about it. For, describe a circumference passing through three consecutive vertices A, B, C (II. 88); let O be its centre, draw OH perpendicular to BC and bisecting it at H, and join OA, OD. Conceive the quadrilateral AOHB to be revolved upon the line OH (i. e., folded over), until HB falls upon its equal HC. The polygon being regular, the angle НВА = HCD, and the side BA = CD; therefore the side BA will take the direction of CD and the point A will fall upon D. Hence OD = OA, and the circumference described with the radius 04 and passing through the three consecutive vertices A, B, C, also passes through the fourth vertex D. It follows that the circumference which passes through the three vertices B, C, D, also passes through the next vertex E, and thus through all the vertices of the polygon. The circle is therefore circumscribed about the polygon. 2d. A circle may be inscribed in it. For, the sides of the polygon being equal chords of the circumscribed circle, are equally distant from the centre; therefore, a circle described with the centre O and the radius OH will touch all the sides, and will consequently be inscribed in the polygon. 6. Definitions. The centre of a regular polygon is the common centre, O, of the circumscribed and inscribed circles. The radius of a regular polygon is the radius, OA, of the circumscribed circle. The apothem is the radius, OH, of the inscribed circle. The angle at the centre is the angle, A OB, formed by radii drawn to the extremities of any side. 7. The angle at the centre is equal to four right angles divided by the number of sides of the polygon. 8. Since the angle ABC is equal to twice ABO, or to ABO+ BAO, it follows that the angle ABC of the polygon is the supplement of the angle at the centre (I. 68). Amor, 18, PROPOSITION III.—THEOREM. 9. Regular polygons of the same number of sides are similar. Let ABCDE, A'B'C'D'E', be regular polygons of the same number of sides; then, they are similar. For, 1st, they are mutually equiangular, since the magnitude of an angle of either polygon depends only on the number of the E B' A B H' H sides (7 and 8), which is the same in both. CE D 2d. The homologous sides are proportional, since the ratio AB: A'B' is the same as the ratio BC: B'C', or CD: C'D', etc. Therefore the polygons fulfill the two conditions of similarity. 10. Corollary. The perimeters of regular polygons of the same number of sides are to each other as the radii of the circumscribed circles, or as the radii of the inscribed circles; and their areas are to each other as the squares of these radii. For, these radii are homologous lines of the similar polygons (III. 43), (IV. 24). PROPOSITION IV.-PROBLEM. 11. To inscribe a square in a given circle. Draw any two diameters AC, BD, perpendicular to each other, and join their extremities by the chords AB, BC, CD, DA; then, ABCD is an inscribed square (II. 12), (II. 59). B 12. Corollary. To circumscribe a square about the circle, draw tangents at the extremities of two perpendicular diameters AC, BD. |