227. Definitions. The radius of a regular polygon is the radius of the circumscribed circle, as OE.

The apothem of a regular polygon is the radius of the inscribed circle, as OF.

The central angle of a regular polygon is the angle between two consecutive radii, as EOD.

228. Cor. I. The central angle of a regular polygon is equal to four right angles divided by the number of sides.

229. Cor. II. The central angle of a regular polygon is the supplement of any angle of the polygon.

4

rt. As + rt. s = 2 rt.

EXERCISES

Find X, the unknown term, in the following proportions : 181. 2:3 = 14: X.

183. 9:x = a: b. 182. 3:54 = 3 : 63.

184. 9: x = x : 25. 185. Write in the form of a proportion 12 x = 72. 186. If mn = pq, write all the possible proportions.

187. Illustrate, by using numbers, the effect of alternation in a proportion. Of inversion, composition, division, the ratio between the sum of the antecedents and the sum of the consequents, etc.

188. In Prop. XXI, if AB, BC, and CA are respectively 16, 12, and 20, what are AD, DC, and BD?

189. In Prop. XXI, if BA:BC = m:n, what is the ratio of the perimeters of the triangle M and N ?

190. In Prop. XXI, if the ratio between the angles A and C equals 3:7, how many degrees are there in each of the acute angles at B?

191. In Prop. XXIII, if AE: ED= 8:5 and EB = 35, what is CE ?

192. Two sides of a triangle are 36 and 20 and the altitude upon the third side is 15. Find the radius of the circumscribed circle.

193. The three sides of a triangle are 10, 12, and 16. Find the bisector of the angle between the first two sides.

PROPOSITION XXVIII

230. Theorem. Chords joining points of equal division on a circumference form a regular inscribed polygon, and tangents at these points form a regular circumscribed polygon.

Tangents at these points form FGHKL, a reg. O

PROPOSITION XXIX

231. Theorem. A regular polygon inscribed in a circle being given, a similar polygon may be circumscribed about the circle.

1

Cons.

Dem.

Bisect the S AB, BC, etc. at F, G, etc.
Draw tangents at F, G, etc.

F and G are points of = division, 1 2 3 4 5 = reg. o

being midway points

reg. and has same 1 2 3 4 5 is similar to ABCDE

number of sides

233. Cor. II. Chords AF, BF, BG, etc., form a regular inscribed polygon of double the number of sides of ABCDE.

Tangents at A, F, B, G, etc., form a regular circumscribed polygon having double the number of sides of 123 45.

234. Cor. III. The sides of ABCDE and 1 2 3 4 5 are parallel each to each.