Proposition 16. Theorem. 245. An angle formed by two chords which intersect within a circle, is measured by one-half the sum of the arcs intercepted between its sides and between its sides produced. Hyp. Let BD, CE be two chords intersecting at A within the BCDE. BAC is measured by (arc BC+ arc DE). To prove But Z BAC = AEB + ZABE. The ext. of a ▲ equals the sum of the opp. int. ≤s (98). ZAEB is measured by arc BC, 1. If arc BC 84° and CAD is a rt. 2, how many degrees are there in the arc DE? 2. The sides of a quadrilateral touch a circle, and the straight lines drawn from the centre of the circle to the vertices cut the circumference in A, B, C, D: show that AC, BD, which intersect inside the circle, are at right angles to each other. Proposition 17. Theorem. 246. An angle formed by two secants which intersect without a circle, is measured by one-half the difference of the intercepted arcs. Hyp. Let AC, AB be two secants intersecting at A without the BCED. ZA is measured by To prove E B .. A is measured by (arc BC - arc DE). Q. E.D. 247. SCH. Prop. 14 may be considered as a special case of Props. 16 and 17 by conceiving AB in (245) and (246) to move parallel to its present position until D reaches E. When D reaches E, the arc DE becomes zero, and BAC becomes an inscribed angle, measured by half its intercepted arc. EXERCISES. 1. If arc BC = 80° and B degrees in the angle A. = 14°, find the number of 2. A, B, C are three points on the circumference of a circle, the bisectors of the angles A, B, C meet in D, and AD produced meets the circle in E: prove that ED EC. 3. If a quadrilateral be described about a circle, the angles at the centre subtended by the opposite sides are supplemental. Proposition 18. Theorem. 248. An angle formed by a tangent and a secant is measured by one-half the difference of the intercepted arcs. E H Hyp. Let AC, AB be a tangent and a secant intersecting at A. To prove A is measured by (arc BHE- arc DE). .. A is measured by (arc BHE-arc DE). Q.E.D. 249. COR. The angle formed by two tangents is measured by one-half the difference of the intercepted arcs. EXERCISES. 1. Two tangents AB, AC are drawn to a circle; D is any point on the circumference outside the triangle ABC: show that the sum of the angles ABD and ACD is constant. 2. If a variable tangent meets two parallel tangents it subtends a right angle at the centre. QUADRILATERALS. Proposition 19. Theorem. 250. If the opposite angles of a quadrilateral are supplementary, the quadrilateral may be inscribed in a circle. Hyp. Let ABCD be a quadrilateral in which ZB+D=2-rt. Zs. To prove the pts. A, B, C, D are in the same . Proof. Through the three pts. A, B, C describe a O. B E D will cut AD, or AD produced, at some other pt. than D. Let E be this pt. Join EC. Because the quadrilateral ABCE is inscribed in a O, .. ABC AEC 2 rt. s. The opp. 48 of an inscribed quad. are supplementary (242). ZABC + ADC 2 rt. S. But (Hyp.) that is, an ext. ▲ of a ▲ = an int. opp. /, which is im possible. (98) ..the circle which passes through A, B, C, must pass through D. Q.E.D. 251. DEF. Points which lie on the circumference of a circle are called concyclic. A cyclic quadrilateral is one which is inscribed in a circle. EXERCISE. If two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals: prove that this perpendicular will bisect the angle between the diagonals, Proposition 20. Theorem. 252. In any quadrilateral circumscribing a circle, the sum of one pair of opposite sides is equal to the sum of the other pair. Hyp. Let ABCD be a quadrilateral circumscribing a O. To prove AB+ CD = AD + BC. Proof. From the centre O draw the radii to the pts. of contact E, F, G, H, and draw OB. Then rt. A OBE=rt. A OBF, (110) B E Similarly, EA = HA, GD = HD, GC = FC. Adding these four equations, we have or O G EBEA+ GD + GC = FB + HA+ HD + FC, ABCD BC + AD. Q.E.D. EXERCISES. 1. The line joining the middle points of two parallel chords of a circle passes through the centre. 2. The chords that join the extremities of two equal arcs of a circle towards the same parts are parallel. 3. The sum of the angles subtended at the centre of a circle by two opposite sides of a circumscribed quadrilateral is equal to two right angles. |