:. ▲ ABG, CBH, CDI, etc., are equal and isosceles. (Why?) .. LG LH = ZI, etc., and Whence = AG: = GB = BH = HC, etc. GH = HI= IK, etc. .. circumscribed polygon FGHIK is regular. (Ax. 2.) Q.E.D. 392. COR. 1. The perimeter of a regular inscribed polygon is less than the perimeter of a regular inscribed polygon of double the number of sides. 393. COR. 2. The perimeter of a regular circumscribed polygon is greater than the perimeter of a regular circumscribed polygon of double the number of sides. Ex. 919. An equilateral polygon inscribed in a circle is regular. Ex. 920. An equilateral polygon circumscribed about a circle is regular if the number of its sides is odd. Ex. 921. An equiangular polygon circumscribed about a circle is regular. PROPOSITION IV. PROBLEM 394. To inscribe a square in a given circle. B D Construction. In the given circle ABC, draw diameters AC and BD perpendicular to each other, and join AB, BC, CD, and DA. Then ABCD is the required square. [The proof is left to the student.] 395. COR. 1. By bisecting the central angles, the arcs AB, BC, etc., will be bisected, and a polygon of eight sides may be inscribed in the circle. By repeating the process, polygons of 16, 32, 2" sides may be constructed. 396. COR. 2. By drawing tangents at A, B, C, and D, a square may be circumscribed about the circle. Ex. 922. To circumscribe an octagon about a given circle. Ex. 923. To construct a regular octagon, having a given side. Ex. 924. The side of an inscribed square is equal to the radius multiplied by V2. Ex. 925. Find the area of a square, if its radius is equal to r. PROPOSITION V. PROBLEM 397. To inscribe a regular hexagon in a given circle. B. E Construction. In the given circle ACD, draw the radius AO. From A as a center, with a radius equal to OA, draw an arc meeting the circumference in B. Draw AB. AB is the side of a regular hexagon. By applying the radius six times as a chord, the regular hexagon ABCDEF is formed. 398. COR. 1. By joining the alternate vertices of an inscribed regular hexagon, an inscribed equilateral triangle is formed. 399. COR. 2. Polygons of 3, 6, 12, 24, etc., sides may be inscribed in and circumscribed about a given circle. Ex. 926. The side of an equilateral triangle is equal to its radius multiplied by √3. Ex. 927. If the radius of a circle is r, the apothem of the inscribed hexagon is A equal to V3. Ex. 928. To circumscribe a regular hexagon about a given circle. B Ex. 929. To construct a regular polygon of twelve sides, having given a side. Ex. 930. In Ex. 929 how many degrees are in the angle at the center? Ex. 931. Find the area of a regular hexagon if its radius is equal to r. Ex. 932. The apothem of an equilateral triangle is equal to one-half its radius. Ex. 933. The area of an inscribed equilateral triangle is equal to onehalf the area of the inscribed regular hexagon. Ex. 934. The areas of triangles inscribed in equal circles are to each other as the products of their three sides. Ex. 935. A square constructed on a diameter of a circle is equivalent to twice the area of the inscribed square. PROPOSITION VI. PROBLEM 400. To inscribe a regular decagon in a given circle. B Construction. In the given circle AB, draw any radius OA, and divide it in extreme and mean ratio, so that Then OC is the side of the required decagon, and by applying it ten times as a chord, a regular decagon ADEF, etc., is formed. But AOAD being isosceles, the similar triangle DAC must or But ..arc AD is 20+ZODA + ≤ DAO = 2 rt. . .. 502 rt. 4, 20= rt. 2, or 1 of 4 rt. 4. of the circumference, and AD is the side of an inscribed regular decagon. Q.E.F. 401. COR. 1. By joining the alternate vertices A, E, G, etc., an inscribed regular pentagon is formed. 402. COR. 2. Polygons of 5, 10, 20, etc., sides may be inscribed in and circumscribed about a given circle. PROPOSITION VII. PROBLEM 403. To construct the side of a regular polygon of fifteen sides inscribed in a given circle. D A B Construction. In the given circle ABD, draw the chord AB equal to the radius, and chord AC equal to the side of the inscribed regular decagon. Then CB is the side of the inscribed regular polygon of fifteen sides. Arc CB6, or of the circumference, and CB is the side of the required polygon. Q.E.F |