288. THEOREM. An angle inscribed in a circle is equal to half the central angle subtended by the same arc. Given the angle ABC inscribed in the circle with the center O, and the central angle AOC subtended by the same arc AC. To prove that Proof. ABCAOC. We distinguish three cases: I. When the center O is on a side of the angle ABC. III. When O is without the angle ABC. 289. COROLLARY 1. Angles inscribed in the same arc, or in equal arcs, of a circle are equal B (Fig. 130). FIG. 130. Case I Why? α Α' C 290. COROLLARY 2. An angle inscribed in a semicircle is a right angle (Fig. 131). 291. COROLLARY 3. The opposite angles of a quadrilateral inscribed in a circle are supplementary (Fig. 132). 1. If the diagonals of a quadrilateral inscribed in a circle are diameters, the quadrilateral is a rectangle. 2. If an angle inscribed in a circle is a right angle, the arc in which it is inscribed is a semicircle. D 3. Two chords of a circle drawn perpendicular to a third chord at its extremities are equal. D' D B B 6. If two opposite angles of a quadrilateral are supplementary, a circle may be circumscribed about the quadrilateral. Prove by reductio ad absurdum : Let B and D be supplementary. Describe a circle through A, B, C and suppose it does not pass through D, but cuts AD, or AD produced, in D'. 293. THEOREM. An angle formed by a tangent and a chord of a circle is equal to any inscribed angle subtended by the arc intercepted between the sides of the angle. Given To prove FIG. 133. the circle with the center O, the tangent AB at P, and the chord PC; also the inscribed angles PDC and PEC, subtended by the arcs PEC and PDC, respectively. that ≤ BPC=▲ PDC, and ≤ CPA = ≤ CEP. Proof. 1. Draw the diameter PF, and join FC. 294. COROLLARY 1. An angle formed by a tangent and a chord of a circle is equal to half the central angle subtended by the arc intercepted between the sides of the angle. 295. COROLLARY 2. The tangents to a circle at the extremities of a chord make equal angles with the chord. A FIG. 134. 296. THEOREM. The oblique angle included by two half lines which meet a circle is equal to half the sum, or half the difference, of the central angles subtended by the arcs intercepted by the sides (produced, if necessary) according as the vertex of the angle is within or without the SUGGESTIONS: In Fig. 135, α = ẞ+y; in Figs. 136, 137, 138, α = ß — y. But ẞ and y are one half the central angles subtended by the intercepted arcs. 1. If an arc of a circle contains 60°, find the angles formed by the tangents drawn at its extremities; also the angles formed by the tangents with the chord joining the extremities of the arc. 2. If two tangents to a circle form an angle of 135°, find the number of degrees in the intercepted arcs. 3. AB, BC, CD, DE are consecutive arcs of a circle each containing 15°; also AP and BP are tangeants meeting in P. Find the size of PAB, PAC, PAD, PAE. 5. If two circles are tangent to each other and two straight lines are drawn through the point of contact meeting one of the circles in A and B and the other in A' and B', then the cords AB and A'B' are parallel. Hint: Draw a common tangent through the point of contact and apply § 293. 6. If a circle is described on the radius of another circle as a diameter, any chord in the latter drawn from the point in which the circles meet is bisected by the former. 7. If two circles intersect and a line is drawn through each point of intersection and terminated by the circles, the chords joining the ends of these segments are parallel. Draw AB. Then ZE is supplementary to / CAB. ..LE=LBAD. Why? Why? In like manner it follows that ▲ F=2 CAB. .. E and F are supplementary. Why? 8. If two circles intersect, any two parallel lines drawn through the points of intersection and terminating in the circles are equal. 9. A circle is described on one of the equal sides of an isosceles triangle as a diameter. Prove that the circle bisects the base. REGULAR POLYGONS 298. A regular polygon. If a polygon is both equilateral and equiangular, it is called a regular polygon. The equilateral triangle and the square are examples. 299. Inscribed and circumscribed polygons. If the vertices of a polygon lie on a circle (Fig. 139, (a)), the polygon is said to be inscribed in the circle, and the circle is said to be circumscribed about the polygon. If the sides of a polygon are tangent to a circle (Fig. 139, (b)), the polygon is said to be circumscribed about the circle, and the circle is said to be inscribed in the polygon. |