210. An angle formed by two secants, two tangents, or a tangent and a secant, and which has its vertex without the circumference, is measured by one-half the concave arc, minus one-half the convex arc. 0 CASE I. Let the angle 0 (Fig. 1) be formed by the two secants O A and O B. We are to prove Draw C B. $ 105 (the exterior 2 of a A is equal to the sum of the two opposite interior é ). By transposing, 20=2ACB-LB, (an inscribed Z is measured by ż the intercepted arc). ..2 O is measured by 1 arc A B § 203 § 203 1 arc C E. CASE II. Let the angle 0 (Fig. 2) be formed by the two tan gents 0 A and O B. We are to prove Draw A B. $ 105 (the exterior 2 of a A is equal to the sum of the two opposite interior &). By transposing, 20 =2 A B -2 0 4 B. But Z A B C is measured by : arc A M B, § 209 (an 2 formed by a tangent and a chord is measured by the in'. cepted arc), and 20 A B is measured by 1 arc A S B. $ 209 .. 2 O is measured by ļ arc A M B – 1 arc A S B. CASE III. Let the angle 0 (Fig. 3) be formed by the tangent OB and the secant 0 A. We are to prove 20 is measured by 1 arc A DS – 1 arc C E S. § 105 (the exterior 2 of a A is equal to the sum of the two opposite interior 6). By transposing, Lo=LACS-LCSO. § 203 $ 209 (being an Z formed by a tangent and a chord). i. 2 O is measured by 1 arc A DS – 1 arc C E S. Q. E. D. 211. Two parallel lines intercept upon the circum. ference equal arcs. Let the two parallel lines C A and B F (Fig. 1), inter cept the arcs C B and A F. We are to prove arc CB = arc A F. Draw A B. $ 68 LA= LB, But the arc C B is double the measure of Z A. and the arc A F is double the measure of Z B. .. arc CB= arc A F. Ax. 6. Q. E. D. 212. SCHOLIUM. Since two parallel lines intercept on tho circumference equal arcs, the two parallel tangents M N and O P (Fig. 2) divide the circumference in two semi-circumferences A C B and A Q B, and the line A B joining the points of contact of the two tangents is a diameter of the circle. 213. If the sum of two arcs be less than a circumference the greater arc is subtended by the greater chord; and conversely, the greater chord subtends the greater arc. B с P gether be less than the circumference, and let chord A B > chord BC. Draw A C. In the A ABC § 203 § 117 (in a A the greater Z has the greater side opposite to it). CONVERSELY : If the chord A B be greater than the chord BC. We are to prove arc AB > arc B C. Нур. . .:.2C> A, $ 118 (in a A the greater side has the greater Z opposite to it). .. arc A B, double the measure of the greater Z C, is greater than the arc B C, double the measure of the less < A. Q. E. D. PROPOSITION XX. THEOREM. 214. If the sum of two arcs be greater than a circumference, the greater arc is subtended by the less chord; and, conversely, the less chord subtends the greater arc. § 213 In the circle BCE let the arcs A EC B and BA EC together be greater than the circumference, and chord AB < chord BC. From the given arcs take the common arc A EC; we have left two arcs, C B and A B, less than a circumference, of which C B is the greater. ... chord C B > chord A B, (when the sum of two arcs is less than a circumference, the greater arc is subtended by the greater chord). ... the chord A B, which subtends the greater arc AEC B, is less than the chord B C, which subtends the less arc B A EC. CONVERSELY : If the chord A B be less than chord BC. arc A ECB > arc B A E C. Arc B C + arc B A E C: the circumference. $ 213 Q. E. D. |