THEOREM I. 320. The sect to a point from the center of a circle is less than, equal to, or greater than, the radius, according as the point is within, on, or without the circle. B A PROOF. If a point is on the circle, the sect drawn to it from the center is a radius, for it is one of the positions of the describing sect. Any point, Q, within the circle lies on some radius, OQR, .: OQ< OR. If S is without the circle, then the sect OS contains a radius OR, OS > OR. .. 321. By 33, Rule of Inversion, a point is within, on, or without the circle according as its sect from the center is less than, equal to, or greater than, the radius. 322. A Secant is a line which passes through two points on the circle. THEOREM II. 323. A secant can meet the circle in only two points. PROOF. By definition, all sects joining the center to points on the circle are equal, but from a point to a line there can be only two equal sects. (155. No more than two equal sects can be drawn from a point to a line.) 324. A Chord is the part of a secant between the two points where it intersects the circle. 325. A Segment of a circle is the figure made by a chord and one of the two arcs into which the chord divides the circle. 326. When two arcs together make an entire circle, each is said to be the B Explement of the other. 327. When two explemental arcs are equal, each is a Semicircle. 2 328. When two explemental arcs are unequal, the lesser is called the Minor Arc, and the greater is called the Major Arc. 329. A segment is called a Major or Minor Segment according as its arc is a major or minor arc. HYPOTHESIS. Let F, G, and H be points on a O. Since, by definition, a center is a point from which all sects to the circle are equal, therefore any center of a circle through F and G is in the perpendicular bisector of FG, and any center of a circle through G and H is in the perpendicular bisector of GH. (183. The locus of the point to which sects from two given points are equal is the perpendicular bisector of the sect joining them.) But these two perpendicular bisectors can intersect in only one point, O FGH has only one center. .. 331. COROLLARY I. The perpendicular bisector of any chord passes through the center, 332. COROLLARY II. To find the center of any given circle, or of any given arc of a circle, draw two non-parallel chords and their perpendicular bisectors. The center is the point where these bisectors intersect. 333. A Diameter is a chord through the center. 334. A diameter is equal to two radii : so all diameters are bisected by the center of the circle, and are equal. THEOREM IV. 335. Circles of equal radii are congruent. P HYPOTHESIS. Two circles of which cand o are the centers, and radius CD = radius OP. CONCLUSION. The circles are congruent. PROOF. Apply one circle to the other so that the center 0 shall coincide with center C, and sect OP fall upon line CD. Then, because OP CD, the point P will coincide with the point D. Then every particular point in the one circle must coincide with some point in the other circle, because of the equality of radii. (321. A point is on the circle when its sect from the center is equal to the radius.) 336. COROLLARY. After being applied, as above, the second circle may be turned about its center; and still it will coincide with the first, though the point P no longer falls upon D. Hence, considering one circle as the trace of the other, A circle can be made to slide along itself by being turned about its center, This fundamental property of this curve allows us to turn any figure connected with the circle about the center without changing its relation to the circle. 337. Circles which have the same center are called Concentric. THEOREM V. 338. Different concentric circles cannot have a point in com mon. A A PROOF. The points of the circle with the lesser radius are all within the larger circle. (321. A point is within the circle if its distance from the center is less than the radius.) 339. FIRST CONTRANOMINAL OF 338. Two different circles with a point in common are not concentric. 340. SECOND CONTRANOMINAL OF 338. Two concentric cir. cles with a point in common coincide. 341. The center of a circle is a Center of Symmetry, the end points of any diameter being corresponding points. This follows at once from the definition of Central Symmetry, and the fundamental property that the circle slides along itself when turned about its center, and so coincides with itself after turning about the center through any angle. The circle is the only closed curve which will slide upon its trace. |