touches the circumference at one point only, and does not intersect it if produced; as MN. chord N 166. A chord is a straight line joining any two points in the circumference; as AB. 167. A central angle is an angle formed by two radii, as LA. 168. Circles having the same center are called concentric. PRELIMINARY THEOREMS 169. All radii of the same circle are equal. (By definition.) 170. A point is within, on, or without a circumference, according as the distance from the center is less than, equal to, or greater than a radius. 171. All diameters of the same circle are equal. 172. Two circles are equal if their radii are equal. (Prove by superposition.) 173. A diameter bisects a circle and its circumference. (Prove by superposition.) Ex. 293. The radii of two concentric circles are 6 inches and 9 inches respectively. If a point is 7 inches from the common center, does it lie within the larger circle ? Within the smaller ? Ex. 294. The distance between the centers of two circles is 4, the radii are 6 and 9 respectively. Does every point of the smaller circle lie within the greater ? 174. In the same circle or in equal circles, equal central angles intercept equal arcs; and, conversely, equal arcs subtend equal central angles. To prove Hyp. In equal O, O and O', 20=20'. arc AB= arc A'B'. Hint. — Prove by superposition. Compare (69). CONVERSELY. — Hyp. In equal O, O and O', arc AB = arc A'B'. 20= 20 To prove Hint. — By superposition. 175. Cor. In the same or in equal circles, the greater of two unequal central angles intercepts the greater arc, and conversely. Ex. 295. Divide a circumference into four equal parts. Ex. 299. If from a point A in a circumference a chord AB and a diameter AC are drawn, a radius parallel to AB bisects arc BC. Ex. 300. If through a point equidistant from two points in the circumference a radius is drawn, the arc between the two points is bisected. Ex. 301. If a secant is parallel to a diameter, the lines intercept equal arcs on the circumference. Ex. 302. Any two parallel secants intercept equal arcs on a circumference. (Ex. 301.) Ex. 303. If the perpendiculars drawn from a point in the circumference upon two radii are equal, the point bisects the arc intercepted by the two radii. Ex. 304. If the line joining the midpoints of two radii is equal to the line joining the midpoints of two other radii, the radii intercept equal arcs respectively. Ex. 305. If the perpendiculars from the center upon two chords are equal, the arcs subtended by the chords are equal. 176. DEF. A polygon is inscribed in a circle, if all its vertices are in the circumference. The circle is then said to be circumscribed about the polygon. PROPOSITION II. THEOREM 177. In the same circle, or equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs. Proof. Draw radii, OA, OB, O'A', O'B'. LAOB= L AO'B', AO = A'O', BO= B'O', (radii of equal ®). (s. a. s. = 8. a. s.). .:. chord AB= chord A'B'. CONVERSELY. — Hyp. In equal , 0 and 0', AB= A'B'. arc AB= arc A'B'. Hint. - Draw radii OA, OB, O'A', O'B', and prove the equality of A OAB and O' A'B'. B Q.E.D. To prove А, Ex. 306. If chords AB, BC, CD, DE are equal, then chords AC, BD, CE are equal. E А Ex. 307. If in the annexed diagram AB=CD, then BC = AD. Ex. 308. If in the same diagram two intersecting chords, AD and BC, are equal, then AB = CD. Ex. 309. The diagonals of an equilateral pentagon inscribed in a circle are equal. B В Ex. 310. The radii drawn to the vertices of an inscribed equilateral hexagon divide the figure into six equilateral triangles. Ex. 311. If two chords bisect each other, the arcs intercepted by the sides of a pair of vertical angles are equal. Ex. 312. If two chords bisect each other, they are diameters. PROPOSITION III. THEOREM 178. A radius perpendicular to a chord bisects the chord and the subtended arc. 179. Cor. A perpendicular bisector of a chord passes through the center of the circle. Ex. 313. If two chords are equal, the perpendiculars from the center upon the chords are equal. Ex. 314. If the perpendiculars from the center upon the sides of an inscribed polygon are equal, the polygon is equilateral. Ex. 315. If from a point without a circle two equal lines are drawn to a circumference, the bisector of the angle they form passes through the center of the circle. Ex. 316. Two points, each equidistant from the ends of a chord, determine a line passing through the center of the circle. |