172. DEF. A circle is a plane closed curve, all of whose points are equally distant from a fixed point in the plane, as ABC (37). ARC 173. DEF. The fixed point in the plane (D) is the center. The length of the circle is called the circumference. An arc is any portion of the circumference, as AB. A semicircumference is half of the circumference. A minor arc is an arc less than a semicircumference; a major arc is an arc greater than a semicircumference. The word arc taken alone generally signifies a minor arc. 174. DEF. A radius is a line join DIAMETER E RADIUS ing the center to any point in the circle, as DC. KB A diameter is a straight line through the center terminated at each end by the circle, as EF. 175. DEF. A chord is a straight line joining any two points in the IA secant is circumference, as GH. a straight line that intersects the circle in two points, as IK. A tan gent is a straight line that touches the circle at one point only, and does not intersect it if produced, N M SECANT -K as MN. This point (L) is called the point of contact or point of tangency. 176. DEF. as LEDC. A central angle is an angle formed by two radii, An angle is said to intercept any arc that is cut off by its sides, and this arc is said to subtend the angle. A chord that joins the ends of an arc is said to subtend this arc; thus chord GH subtends the arc GH. 177. DEF. Circles having the same center are called concentric. PRELIMINARY THEOREMS 178. All radii of the same circle are equal. (By definition.) 179. 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. 180. All diameters of the same circle or of equal circles are equal. 181. Two circles are equal if their radii are equal. (Prove by superposition.) 182. A diameter bisects the circumference. (Prove by superposition.) PROPOSITION I. THEOREM 183. In the same circle or in equal circles, equal central angles intercept equal arcs; and, conversely, equal arcs subtend equal central angles. To prove AB = A'B'. Proof. Place the circle whose center is O on the circle whose center is o' so that OA coincides with OA'. Then OB takes the direction OB B coincides with B', .. AB coincides with AB (19) (<0 = 20') (OB = O'B') Otherwise the radius of one would be greater than that of the other. .. AB= AB'. II. CONVERSELY. I. Given in equal , o and o', Q. E. D. HINT. By superposition. 184. COR. In the same or in equal circles, the greater of two unequal central angles intercepts the greater arc, and conversely. 185. METHOD XIII. The equality of arcs in the same circle or equal circles can be demonstrated by the equality of the subtended central angles. Ex. 485. If AB and CD are diameters of the same circle, arc AC arc BD. = Ex. 486. If, in the diagram opposite, AB is a diameter, OD a radius, AC a chord, and Z BOD = 2(ZA), then BD = BC. Ex. 487. If a diameter AB bisects the angle A, which is formed by two chords AC and AD, then BC = BD. Ex. 488. If from a point A in a circumference a chord AB and a diameter AC are drawn, a radius parallel to AB bisects BC. Ex. 489. If through a point equidistant from two points in the circumference a radius is drawn, the arc between the two points is bisected. Ex. 490. If a secant is parallel to a diameter, the lines intercept equal arcs on the circumference. Ex. 491. Any two parallel secants intercept equal arcs on a circumference. (Ex. 490.) Ex. 492. 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. 493. If the line joining the mid-points of two radii is equal to the line joining the mid-points of two other radii, the radii intercept equal arcs respectively. Ex. 494. Divide a circumference into four equal parts. 186. The circumference of a circle may be divided into 360 equal parts called degrees. A degre may be subdivided into 60 equal parts called minutes, and similarly a minute may be divided into 60 seconds. An angle is measured by an arc, if both angle and arc contain the same number of degrees. Ex. 497. Prove that a central angle is measured by its intercepted arc if the central angle equals Ex. 498. Construct an arc of (a) 45o, (b) 60°, (c) 150°. ers). 187. DEF. A polygon is inscribed in a circle, if all its vertices are in the circle; as ABCDE. The circle is then said to be circumscribed about the polygon. E The center of the circumscribed circle is Α called the circumcenter of the polygon. PROPOSITION II. THEOREM 188. In the same circle, or in equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs. |