PROPOSITION XVII. THEOREM. 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. Let the angle 0 (Fig. 1) be formed by the two secants OA and O B. We are to prove 20 is measured by arc A B arc EC. Draw CB. = ZACB Z 0 + 2 B, § 105 (the exterior of a ▲ is equal to the sum of the two opposite interior ▲). By transposing, 20=ZACB-ZB, ZACB is measured by arc A B, But § 203 and ZB is measured by arc CE, § 203 .. ≤ 0 is measured by arc A B - 1 arc C E. CASE II. Let the angle 0 (Fig. 2) be formed by the two tangents OA and O B. We are to prove O is measured by 1⁄2 arc A M B – 1⁄2 arc A S B. Draw A B. = LABC 20+ 20AB, § 105 (the exterior of a ▲ is equal to the sum of the two opposite interior ≤). By transposing, But 40=ZABC-ZO AB. ZABC is measured by arc AMB, § 209 § 209 (an formed by a tangent and a chord is measured by the incepted arc), and ZOAB is measured by arc A S B. ..ZO is measured by arc AMB arc ASB. CASE III. Let the angle 0 (Fig. 3) be formed by the tangent OB and the secant O A. We are to prove 20 is measured by arc A DS-arc CE S. Draw C S. § 105 (the exterior of a ▲ is equal to the sum of the two opposite interior 4). SUPPLEMENTARY PROPOSITIONS. PROPOSITION XVIII. THEOREM. 211. Two parallel lines intercept upon the circum ference equal arcs. Let the two parallel lines CA and B F (Fig. 1), inter But the arc CB is double the measure of ▲ A. and the arc AF is double the measure of ≤ B. .. arc C Barc A F. Ax. 6. Q. E. D. 212. SCHOLIUM. Since two parallel lines intercept on the circumference equal arcs, the two parallel tangents MN and OP (Fig. 2) divide the circumference in two semi-circumferences ACB and AQ B, and the line A B joining the points of contact of the two tangents is a diameter of the circle. PROPOSITION XIX. THEOREM. 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 C P In the circle ACP let the two arcs A B and BC together be less than the circumference, and let AB be the greater. C, measured by the greater arc A B, is greater than A, measured by the less arc BC. .. the side A B > the side BC, § 203 $ 117 (in a ▲ the greater has the greater side opposite to it). CONVERSELY: If the chord AB be greater than the (in a ▲ the greater side has the greater Z opposite to it). .. arc A B, double the measure of the greater than the arc B C, double the measure of the less A. C, is greater Q. E. D. 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. B A E In the circle BCE let the arcs AECB and BAEC together be greater than the circumference, and let arc AEC B be greater than arc BAE C. We are to prove chord A B< chord B C. From the given arcs take the common arc A E C; we have left two arcs, CB and A B, less than a circumference, of which CB is the greater. .. chord C B > chord A B, $ 213 (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 BC, which subtends the less arc BAE C. CONVERSELY: If the chord A B be less than chord B C. |