Plane and solid geometry . = chord CD, and let OE andOF be the distances of AB and CD from center 0, respectively. To prove OE = OF. Argument 1. Draw radii OB and OC, 2. E and F are the mid-points of AB and CDy respectively. 3. . rt. A OEB and OCF, EB = CF. 4. OB = OC, 5. .-. A OEB = A OCF. 6. .-. 0E= OF, II. Conversely: Given circle 0 with OE, the distance of chord AB from center0, equal to OF, the distance of chord CD from center 0. To prove chord AB = chord CD,Hint. Prove A OEB = A OCF. Reasons 1. § 54, 15. 2. § 302. 3. § 54, 8 a. 4. § 279, a, 5. § 211. 6. § 110. Ex. 431. If perp
Plane and solid geometry . = chord CD, and let OE andOF be the distances of AB and CD from center 0, respectively. To prove OE = OF. Argument 1. Draw radii OB and OC, 2. E and F are the mid-points of AB and CDy respectively. 3. . rt. A OEB and OCF, EB = CF. 4. OB = OC, 5. .-. A OEB = A OCF. 6. .-. 0E= OF, II. Conversely: Given circle 0 with OE, the distance of chord AB from center0, equal to OF, the distance of chord CD from center 0. To prove chord AB = chord CD,Hint. Prove A OEB = A OCF. Reasons 1. § 54, 15. 2. § 302. 3. § 54, 8 a. 4. § 279, a, 5. § 211. 6. § 110. Ex. 431. If perpendiculars from the center of a circle to the sides ofan inscribed polygon are equal, the polygon is equilateral. Ex. 432. If through any point in a diameter two chords are diawnmaking equal angles with the diameter, the two chords are equal. 124 PLANE GEOMETRY Proposition VIII. Theorem 308. Lv equal circles, or in tlve same circle, if twochords are unequal, the greater chord is at the less dis-tance from the Given circle 0 with chord AB > chord CD, and let OF andOH be the distances of AB and CB from center O, respectively. To prove OF CD. 5. .-. .4D > ^J?;. 6. i^ and C are the mid-points of AB and AEy AF:> Zl > ^/O = Z OCX 7. 8. .-. Z3 < OF < = OH,.-. 0F< OH. 1. § 54, 15. 2. § 155. 3. § 54, 15. 4. § By hyp 5. §309. 6. §302. 7. § 54, 8 b. 8. §166. 9, § 64. 10. § 54, 6. 11. § 164. 12. § 307, I. 13. § 309. 309. Note. The student should give the full statement of the sub-stitution made; thus, reason 5 above should be : ** Substituting AE forits equal CD: BOOK n 125 Proposition IX. Theorem (Converse of Prop. VIII) 310. In equal circles, or in the same circle, if two chordsare ztnequally distant froin the center, tJie chord at theledS distance is the greater.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
