PROPOSITION IV. PROBLEM 180. To circumscribe a circle about a given triangle. Required. To circumscribe a circle about ▲ ABC. AC. Erect perpendicular-bisectors upon BC and Construction. They will intersect at some point E. (Why?) From E as a center, with a radius equal to EA, describe a O. It will pass through A, B, and C. .. ABC is the required O. (Why?) 181. COR. 1. Three points not in a straight line determine a circumference. 182. COR. 2. A circumference cannot be drawn through three points which lie in the same straight line. 183. COR. 3. A straight line cannot intersect a circumference in more than two points. 184. COR. 4. Two circumferences cannot meet in more than two points. Ex. 317. To find the center of a given circle. PROPOSITION V. THEOREM 185. In the same or in equal circles, equal chords are equally distant from the center; and, conversely, chords equally distant from the center are equal. HINT. What is the means of proving the equality of lines? To prove OE = OH, OE LAB, OH1 CD. AB = CD. The proof is similar to the above. 186. REMARK. The equality of two chords is usually established by means of equal distances from the center or equal subtended arcs. Ex. 319. If from any point in the circumference two chords are drawn making equal angles with the radius to the point, these chords are equal. Ex. 320. If through any point in a radius two chords are drawn making equal angles with the radius, these chords are equal. Ex. 321. A line joining the point of intersection of two equal chords to the center bisects the angle formed by the chords. Ex. 322. In a given circle, to draw a chord equal and parallel to a given chord. PROPOSITION VI. THEOREM 187. In the same or in equal circles, the greater of two minor arcs is subtended by a greater chord; and, conversely, the greater chord subtends the greater arc. Proof. Draw radii OA, OB, O'A', O'B'. CONVERSELY. Hyp. In equal , O and O', B (175) (128) Q.E.D. (Hyp.) Ex. 323. In a given circle, to draw a chord equal and perpendicular to a given chord. Ex. 324. In a given circle, to draw a chord equal to a given chord, and parallel to a given line. Ex. 325. In a given circle, to draw a chord equal to one-half a given chord, and perpendicular to a given line. 188. In the same circle, or in equal circles, chords unequally distant from the center are unequal, the nearer one being the greater; and, conversely. Hyp. In ABDC, OE chord AB, OF chord CD, and Since OG OF, G lies between 0 and F. .. KH lies between 0 and CD. 189. COR. The diameter is greater than any other chord. 190. REMARK. - The inequality of chords is usually established by means of unequal distances from the center or by means of unequal arcs. NOTE. The following table will be found convenient. Prop. I, II, V, VI, VII, and several others, may be represented by the following schedule: If, e.g. chords are equal, then central angles, minor arcs, etc., are equal; similarly for unequal parts. [DEF. A segment of a circle is a portion of a circle bounded by a chord and its arc. A sector of a circle is a portion of a circle bounded by two radii and their intercepted arc.] |