Plane and solid geometry . I. Given circle 0, with chord AB > chord CD,To prove AB > CB, Argument 1. Draw radii OA^ OB, OC, OD, 2. In A OAB and OCD, OA = OC, OB = OD. 3. Chord AB > chord CD. 4. .-. Zl > Z2, 5. ,\ AB > CD. II. Conversely: Given circle 0, with AB > prove chord AB > chord CD. Reasoxs 1. § 54, 15. 2. § 279, a. 3. By hyp. 4. § 173. 5. § 294. Argument 1. Draw radii OA, OB, OC, OD. 2. In A OAB and OCD, OA = OC, OB = OD. 3. AB > CD. 4. .-. Zl > Z 2. 5. .-. chord AB > chord CD. Ex. 422. Prove the converse of Prop. IV by the indirect met
Plane and solid geometry . I. Given circle 0, with chord AB > chord CD,To prove AB > CB, Argument 1. Draw radii OA^ OB, OC, OD, 2. In A OAB and OCD, OA = OC, OB = OD. 3. Chord AB > chord CD. 4. .-. Zl > Z2, 5. ,\ AB > CD. II. Conversely: Given circle 0, with AB > prove chord AB > chord CD. Reasoxs 1. § 54, 15. 2. § 279, a. 3. By hyp. 4. § 173. 5. § 294. Argument 1. Draw radii OA, OB, OC, OD. 2. In A OAB and OCD, OA = OC, OB = OD. 3. AB > CD. 4. .-. Zl > Z 2. 5. .-. chord AB > chord CD. Ex. 422. Prove the converse of Prop. IV by the indirect method Reasoxs 1. § 54, 15. 2. § 279, a. 3. By hyp. 4. § 294. 5. § 172. BOOK II 121 Proposition V. Theorem 302. The diameter perpendicular to a chord bisectsthe chojd and also its subtended Given chord AB and diameter CD _L AB at prove AE = EB, AG = CB, and AB = DB, Argument 1. Draw radii OA and OB. 2. In A OAB, OA = OB, 3. . OAB is an isosceles A. 4. /. OE bisects AB, and AE = EB, 5. Also OE bisects Z BOA, and Zl = Z2. 6. .-. Zaod = Zdob. 7. .-. AC= CB and AD = DB. Reason^ 1. § 54, 15. 2. § 279, a. 3. § 94. 4. § 212. 5. § 212. 6. § 75. 7. § 293, I. 303. Cor. I. The perpendicular bisector of a chordparses through the center of the circle. 304. Cor. II. The locus of the centers of all circleswhich pass through two given points is the perpendicu-lar bisector of tlxe line which joins the points. 305. Cor. m. The locus of the inid-points of all chordsof a circle parallel to a given line is the diameter per-pendicular to the line. (For complete proof, see p. 298.) Ex. 423. If the diagonals of an inscribed quadrilateral are unequal, itsopposite sides are unequal. 122 PLANE GEOMETRY Ex. 424. Through a given point within a circle construct a chordwhich shal
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