PROBLEM II. 338. To inscribe a regular hexagon in a circle. Draw any radius, as OA, and with A as a centre and the radius of the circle describe an arc, cutting the circumference at B. .. Draw AB and OB. Now, the ABO is both equilateral and equiangular; (97) arc AB = a of 2 Ls of 4 Ls; = = (81) of the circumference, and the chord AB is the side of a regular inscribed hexagon; ... ABCDEF, which is formed by applying the radius six times as a chord, is the required hexagon. Q. E. F. 339. COR. 1.—To inscribe an equilateral triangle, join the alternate vertices of a regular inscribed hexagon. 340. COR. 2.-To inscribe a regular polygon of 12 sides, bisect the arcs subtended by the sides of a regular inscribed hexagon and draw chords; and by continuing the process, we can inscribe regular polygons of 24, 48, etc., sides. PROBLEM III. 341. In a given circle to inscribe a regular decagon. Suppose the problem to be solved, and let ABC, etc., be the regular inscribed decagon. Draw AC and BD. Now, AC and BD bisect the circumference; they are diameters and intersect at the centre C. Draw BE, cutting AC at P. La is measured by (arc AB + arc EC), or arc BC, and (212) Lb is measured by arc BC; (206) A APB is isosceles, and A B = BP. Also, d is measured by arc ED, or arc AB, and Le is measured by arc AB; (205) A BOP is isosceles, and OP = BP = AB. Lc is measured by arc AE, or AB; (206) ... As APB and ABO are mutually equiangular and similar; or AO : AB :: AB : AP, AO OP :: OP: AP. (269) But this shows that AO, the radius, is divided in extreme and mean ratio at P, and that OP, the greater part, equals AB, a side of the regular inscribed decagon. Therefore, to inscribe a regular decagon, divide the radius in extreme and mean ratio, and apply the greater part ten times as a chord. Q. E. F. 342. COR. 1.-To inscribe a regular pentagon, join the alternate vertices of a regular inscribed decagon. 343. COR.-To inscribe a regular polygon of 20 sides, bisect the arcs subtended by the sides of a regular inscribed decagon and draw chords; and by continuing the process, we can inscribe regular polygons of 40, 80, etc., sides. PROBLEM IV. 344. In a given circle to inscribe a regular pentadecagon. Let C be the given O. C B D Draw the chord AB equal to the side of a regular inscribed hexagon, and the chord BD equal to the side of a regular inscribed decagon, and draw AD. Therefore chord AD = a side of a regular inscribed pentadecagon; and hence if we apply AD fifteen times as a chord we get the required polygon. Q. E. F. 345. COR.-To inscribe a regular polygon of 30 sides, bisect the arcs subtended by the sides of a regular inscribed pentadecagon and draw chords; and by continuing the process, we can inscribe regular polygons of 60, 120, etc., sides. |