. Elements of plane and spherical trigonometry . is drawn perpendicular to BC, cutting CKin L ; show that a j- _ ab sin Ca-\-b cos C 57. Two circles, whose centres are at a distance of 76 from eachother, lie in the same plane but do not intersect. Their externalcommon tangent meets the line of centres produced at an angleof 15°, and their internal common tangent meets the same line atan angle of 37°. Find the radii of the circles. Generalize theproblem. 58. Two circles, whose radii are 53 and 31, and the line joiningwhose centres is 72, are tangent to a third circle whose radius is the
. Elements of plane and spherical trigonometry . is drawn perpendicular to BC, cutting CKin L ; show that a j- _ ab sin Ca-\-b cos C 57. Two circles, whose centres are at a distance of 76 from eachother, lie in the same plane but do not intersect. Their externalcommon tangent meets the line of centres produced at an angleof 15°, and their internal common tangent meets the same line atan angle of 37°. Find the radii of the circles. Generalize theproblem. 58. Two circles, whose radii are 53 and 31, and the line joiningwhose centres is 72, are tangent to a third circle whose radius is the angle between the line of centres of the first circles andthe line through their points of tangency with the third circle. CHAPTEE VII. MISCELLANEOUS PROBLEMS AND TRIGONOMETRIC EQUATIONS. 68. The Radius of the Circle inscribed in anyTriangle. Let r = the radius required. Fig. 28. We have firstAE=AF=x, BF=BD = y, CB= CE = &(a? + y + *) = a + 6 + c = &,or x + y + z = s. Then x = s — (y + z) = s — a, y = S—(z + x)=S — b,. (92) z = s — (x + y) = s — r = ^ tan J J. == .Bi) tan %B, etc., r = (s — a) tan J A = (s — b) tan %B, for tan %A we substitute its value taken from (87), wehave r = (s v «)a/(5- -b)(s- ?c) a)\ s (s — a) (s — a) (s - -b)(s — <0 (93) If the values of tan iA, tan %B, and tan JO be taken from(87) and multiplied together, we have tan IA tan IB tan iC= — \ s \ s (5 — a) (3 — 6) (5 — c) r = s tan JJ. tan $2? tan J (7. (94) 97 98 PLANE TRIGONOMETRY. Fig. 29. 69. The Radius of the Circle circumscribed aboutany Triangle. Let B = the radius 0 the centre of the circum-scribed circle draw OB = B and OEbisecting BC in D. This line alsobisects the arc BEG at E. Hence thearc BE measures the angle A. There-fore BOE=A. We have then fromthe right triangle BOB BD= OB sin BOB,or ia = B sin A, and similarly ib = B sin B, \c = B sin C,adding i(a-\-b-\-c) = s = R (sin A -f sin B -f- sin C)but, § 60, Exercise 13
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