BOOK V. REGULAR POLYGONS AND CIRCLES. 363. DEF. A Regular Polygon is a polygon which is equilateral and equiangular. PROPOSITION I. THEOREM. 364. Every equilateral polygon inscribed in a circle is a regular polygon. C D B Let ABC, etc., be an equilateral polygon inscribed in a circle. We are to prove the polygon A B C, etc., regular. The arcs A B, BC, C D, etc., are equal, § 182 .. arcs A B C, BC D, etc., are equal, Ax. 6 .. the A, B, C, etc., are equal, (being inscribed in equal segments). .. the polygon ABC, etc., is a regular polygon, being equilateral and equiangular. Q. E. D. PROPOSITION II. THEOREM. 365. I. A circle may be circumscribed about a regular polygon. II. A circle may be inscribed in a regular polygon. regular polygon, and also a ○ may be inscribed in this regular polygon. CASE I. Describe a circumference passing through A, B, and C. From the centre 0, draw 0 A, OD, and draw Os to chord BC. On Os as an axis revolve the quadrilateral O A B 8, until it comes into the plane of O s C D. The line s B will fall upon s C, (for LOS B = 40s C, both being rt. 4). The point B will fall upon C, (since 8 B = s The line BA will fall upon CD, (since ▲ B = 4 C, being of a regular polygon). (since В A = CD, being sides of a regular polygon). .. the line OA will coincide with line O D, (their extremities being the same points). ..the circumference will pass through D. § 183 $ 363 § 363 In like manner we may prove that the circumference, passing through vertices B, C, and D will also pass through the vertex E, and thus through all the vertices of the polygon in succession. CASE II.The sides of the regular polygon, being equal chords of the circumscribed O, are equally distant from the centre, $ 185 .. a circle described with the centre O and a radius Os will touch all the sides, and be inscribed in the polygon. § 174 Q. E. D. 366. DEF. The Centre of a regular polygon is the common centre of the circumscribed and inscribed circles. 367. DEF. The Radius of a regular polygon is the radius. OA of the circumscribed circle. 368. DEF. The Apothem of a regular polygon is the radius Os of the inscribed circle. 369. DEF. The Angle at the centre is the angle included by the radii drawn to the extremities of any side. PROPOSITION III. THEOREM. 370. Each angle at the centre of a regular polygon is equal to four right angles divided by the number of sides of the polygon. A D B Let ABC, etc., be a regular polygon of n sides. 371. COROLLARY. The radius drawn to any vertex of a regular polygon bisects the angle at that vertex. 372. Two regular polygons of the same number of sides Let Q and Q' be two regular polygons, each having n sides. We are to prove Q and Q' similar polygons. The sum of the interior of each polygon is equal to 2 rt. 4 (n − 2), $ 157 (the sum of the interior of a polygon is equal to 2 rt. & taken as many times less 2 as the polygon has sides). (for the of a regular polygon are all equal, and hence each to the sum of the divided by their number). is equal Also, each of Q' 2 rt. (n-2) = n § 158 .. the two polygons Q and Q' are mutually equiangular. ..the two polygons have their homologous sides proportional; .. the two polygons are similar. $ 278 Q. E. D. PROPOSITION V. THEOREM. 373. The homologous sides of similar regular polygons have the same ratio as the radii of their circumscribed circles, and, also as the radii of their inscribed circles. Let 0 and O' be the centres of the two similar regular polygons ABC, etc., and A'B'C', etc. From O and O' draw O E, OD, O' E', O' D', also the Is Om and O'm'. O E and O' E' are radii of the circumscribed, and Om and O'm' are radii of the inscribed . $ 367 § 368 the OED, ODE, O' E' D' and O' D' E' are equal, § 371 (being halves of the equal & FED, EDC, F' E' D' and E' D' C'); .. the AO ED and O' E' D' are similar, $280 (if two have two of the one equal respectively to two of the other, they are similar). ED = E' D' OE (the homologous sides of similar are proportional). |