BOOK V REGULAR POLYGONS. MEASUREMENT OF THE CIRCLE REGULAR POLYGONS PROPOSITION I. THEOREM 388. DEF. A regular polygon is a polygon which is both equiangular and equilateral. 389. A circle can be circumscribed about any regular polygon. Given ABCDE, a regular polygon of n sides. To prove that a circle can be circumscribed about ABCDE. Proof. Construct a circle through A, B, and C, and let o be its center. In like manner, it may be proved that the circle passes through the remaining vertices of the polygon. .. a circle can be circumscribed about the given polygon. PROPOSITION II. THEOREM Q. E. D. 390. A circle can be inscribed in any regular polygon. HINT. Circumscribe a circle about the given polygon and prove that the center is equidistant from the sides. 391. DEF. The center (0) of a regular polygon is the common center of the circumscribed and inscribed circles of the polygon. 392. DEF. The radius of a E regular polygon is the radius of the circumscribed circle; as OA. 393. DEF. The central angle is the angle between two radii drawn to the ends of one side; as AOB. B (197) 394. DEF. The apothem of the polygon is the radius of the inscribed circle; as OF. 395. COR. The central angle of a regular polygon of n sides is equal to right angles. 4 Ex. 1282. If ABCDEF is a regular hexagon, then LADCLAEC LAFC. = Ex. 1283. If two diagonals AC and BE of a regular polygon ABCDEFG intersect in Q, then AQ × QC = BQ × QE. Ex. 1284. If in the regular heptagon ABCDEFG, the prolongations of AB and the diagonal of EC meet in H, then HB × HA : = HC × HE. Ex. 1285. Any central angle of a regular polygon is the supplement of an angle of the polygon. Ex. 1286. A triangle is regular if the centers of the circumscribed and inscribed circles coincide. Ex. 1287. A polygon is regular if the centers of the circumscribed and inscribed circles coincide. 396. If the circumference of a circle is divided into any number of equal parts: (1) The chords joining the points of division succes sively form a regular inscribed polygon. (2) Tangents drawn at the points of division form a regular circumscribed polygon. (2) To prove tangents drawn at A, B, C, etc., form the regular circumscribed polygon FGHIK. (Why?) (Why?) .. ▲ ABG, CBH, CDI, etc., are congruent and isosceles. (Why?) and Whence .. ≤ G = ≤ H = ▲ I, etc. AG = GB = BH = HC, etc. .. circumscribed polygon FGHIK is regular. (Ax. 2.) Q. E. D. 397. 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. 398. 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. 1288. If a regular hexagon is inscribed in a circle, and the midpoints of the arcs which are subtended by the sides are joined to the vertices of the hexagon, a regular polygon of twelve sides is formed. Ex. 1289. An equilateral polygon inscribed in a circle is regular. Ex. 1290. An equiangular polygon circumscribed about a circle is regular. PROPOSITION IV. PROBLEM 399. To inscribe a square in a given circle. Construction. In the given circle ABC, draw diameters AC and BD perpendicular to each other, and draw AB, BC, CD, and DA. Then ABCD is the required square. [The proof is left to the student.] 400. 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 be constructed. 16, 32, ***, B 2" sides may |