13. a (b2+c2)cos A+b(c2+a2) cos B+c(a2+b2) cos C=3 abc. 17. a+b+c= (b + c) cos A + (c + a) cos B + (a + b) cos C. 18. b+c-a (b+c) cos A - (ca) cos B+ (a - b) cos C. = 19. a cos (A+B+C) - b cos (B+A) - c cos (A+ C) = 0. 28. tan B÷tan C = (a2 + b2 — c2) ÷ (a2 − b2 + c2). 29. a2 + b2 + c2 = 2(ab cos C + be cos A + ca cos B). 32. If p is the length of the perpendicular from A on 37. If sin B + sin2C = sin A, then A = 90°. 38. If D is the middle point of BC, prove that 39. If a 2b, and A = 3B, prove that C = 60°. = 40. If D, E, F, are the middle points of the sides, BC, CA, AB, prove 4(AD2 + BE2 + CF2) = 3 (a2 + b2 + c2). 41. If a, b, c, the sides of a triangle, are in arithmetic progression, prove A tan tan c - b C 1 , prove that A = 60°. с 2 sin C' prove that B = C. 44. If a2= b2bc+c2, prove that A = 60°. 45. If the sides of a triangle are a, b, and Va2 + ab + b2, prove that its greatest angle is 120°. 46. Prove that the vertical angle of any triangle is divided by the median which bisects the base, into segments whose sines are inversely proportional to the adjacent sides. 47. If AD be the median that bisects BC, prove (1) (b2 - c2) tan ADB = 2 bc sin A, and (2) cot BAD +cot DAC = 4 cot A + cot B + cot C. 48. Find the area of the triangle ABC when a = 625, b= 505, c = 904 yards. Ans. 151872 sq. yards. 49. Find the radii of the inscribed and each of the escribed circles of the triangle ABC when a = 13, b = 14, c = 15. Ans. 4; 10.5; 12; 14. 50. Prove the area S = a2 sin B sin C cosec A. 53. Prove that the lengths of the sides of the pedal triangle, that is, the triangle formed by joining the feet of the perpendiculars, are a cos A, b cos B, c cos C, respectively. 54. Prove that the angles of the pedal triangle are, respectively, — 2 A, π − 2 В, π- 2 C. 57. Prove that the area of the incircle: area of the tri Prove the following statements : 58. If a, b, c, are in A. P., then ac=6rR. 59. If the altitude of an isosceles triangle is equal to the base, R is five-eighths of the base. 60. bc = 4 R2 (cos A+ cos B cos C). 61. If C is a right angle, 2r+2R = a + b. 62. 723 +371 + 717°2 = 82. 66. If P1, P2, P3 be the distances to the sides from the circumcentre, then 71. The distances between the centres of the inscribed and escribed circles of the triangle ABC are 4R sin 2' 73. In an equilateral triangle 3 R = 6r = 2 r1. 74. If r, 71, 72, 73 denote the radii of the inscribed and escribed circles of a triangle, 75. The sides of a triangle are in arithmetic progression, and its area is to that of an equilateral triangle of the same perimeter as 3 is to 5. Find the ratio of the sides and the value of the largest angle. Ans. As 7, 5, 3; 120°. 76. If an equilateral triangle be described with its angular points on the sides of a given right isosceles triangle, and one side parallel to the hypotenuse, its area will be 2 a2 sin2 15° sin 60°, where a is a side of the given triangle. 77. If h be the difference between the sides containing the right angle of a right triangle, and S its area, the diameter of the circumscribing circle = √h2+4S. prove 78. Three circles touch one another externally that the square of the area of the triangle formed by joining their centres is equal to the product of the sum and product of their radii. 79. On the sides of any triangle equilateral triangles are described externally, and their centres are joined: prove that the triangle thus formed is equilateral. 80. If 01, 02, O3 are the centres of the escribed circles of a triangle, then the area of the triangle 0,0,0, = area of a + c - b a+b -c 81. If the centres of the three escribed circles of a triangle are joined, then the area of the triangle thus formed is where is the radius of the inscribed circle of the abc 2 r original triangle. |