BOOK V

REGULAR POLYGONS AND THE CIRCLE

RATIOS AND MEASUREMENT

Topics for Definition.

197. Regular polygon.

198. Polygon inscribed in a circle.

Circle circumscribed about a polygon.

199. Polygon circumscribed about a polygon.

Circle inscribed in a polygon.

THEOREM I

200. An equilateral polygon inscribed in a circle is a regular polygon.

NOTE.

For convenience in constructing this figure, take the length of the radius as a side of the polygon.

Hyp. Let the equilateral polygon ABCDEF be inscribed in the OS.

ABC, BCD,

To prove that polygon A-F is equiangular and therefore regular. SUG. Prove that the arcs intercepted by the etc., are equal. Then apply Bk. II, Theo. XVII. Conclusion.

THEOREM II

201. A circle may be circumscribed about, or inscribed in, a regular polygon.

Hyp. Let ABCDEF be a regular polygon.

To prove that a circle may be circumscribed about, or inscribed in, the polygon A-F.

SUG. 1. Find a pt. O equidistant from A, B, and C. Auth. Draw AO, BO, CO, DO, etc.

SUG. 2. Prove ▲ ABO and BCO equal and isos. Auth. .. LABO= Z OBC = LBCO.

=

SUG. 3. ABO = B. Why? .. ≤ OCBC. Why? SUG. 4. OCDC=2OBC. ABCOA CDO. Auth. .. ODOC.

SUG. 5. In same way prove E and F are the same distance from O as A, B, C, and D. First conclusion.

SUG. 6. Draw OX1 AB, OY1 BC, OZ 1 CD, etc., and apply Bk. II, Theo. X. Second conclusion.

203. Cor. The angle at the centre of a regular polygon is equal to four right angles divided by the number of sides.

204. Problem XXII. To inscribe a regular hexagon in a circle. SUG. See figure under Theo. I.

205. Problem XXIII. To inscribe a square in a circle.

EXERCISES

134. To inscribe a regular triangle or a regular dodecagon in a circle.

SUG. See Prob. XXII.

135. To inscribe a regular octagon in a circle.

SUG. See Prob. XXIII.

136. Find the angle at the centre of a regular pentagon.

137. Of a regular hexagon.

138. Of a regular octagon.

139. Of a regular decagon.

THEOREM III

206. If the circumference of a circle be divided into any number of equal arcs, the chords subtending these arcs form a regular inscribed polygon; the tangents drawn at the points of division form a regular circumscribed polygon.

PART I. Hyp. In the OS, let the equal & AB, BC, CD, DE, and EA be subtended by the chords AB, BC, CD, etc.

To prove that the inscribed polygon A-E is regular.
SUG.

Consult Theo. I.

First conclusion.

PART II. Hyp. At pts. A, B, C, D, and E, on the circumference of OS, let tangents be drawn intersecting at F, G, H, K, and M.

To prove that circumscribed polygon F-M is regular.

SUG. 1. Prove ▲ AFB, BGC, etc., to be equal and isos. A, using Bk. II, Theo. XVIII.

SUG. 2. From Sug. 1, ≤F=≤ G = ZH, etc., and FB + BG = GC + CH, etc.

=

Second conclusion.

207. Cor. 1. If the vertices of a regular inscribed polygon are connected with the middle points of the arcs subtended by the sides of the polygon, a regular inscribed polygon of double the number of sides is formed.

208. Cor. 2. If a regular polygon be circumscribed about a circle, and if at the middle points of the arcs between the points of tangency tangents are drawn, a regular circumscribed polygon of double the number of sides is formed.

EXERCISES

140. Find the value of an angle at a vertex of a regular hexagon; of a regular octagon; of a regular decagon; of a regular dodecagon.

141. A central angle of any regular polygon is the supplement of an angle of the polygon.