# 17. Properties of Triangles’ Problems¶

1. The sides of a triangle are $8$ cm, $10$ cm and $12$ cm. Prove that the greatest angle is double the smallest angle.

2. In a $\triangle ABC,$ if $\frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13},$ prove that $\frac{\cos A}{7} = \frac{\cos B}{19} = \frac{\cos C}{25}$

3. If $\triangle = a^2 - (b - c)^2,$ where $\triangle$ is the area of the $\triangle ABC,$ then prove that $\tan A = \frac{8}{15}$

4. In a triangle $ABC,$ the angles $A, B, C$ are in A.P. Prove that $2\cos\frac{A - C}{2} = \frac{a + c}{\sqrt{a^2 - ac + c^2}}$

5. If $p_1, p_2, p_3$ be the altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively and $\Delta$ be the area of the triangle, prove that $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} = \frac{2ab\cos^2\frac{C}{2}}{\Delta(a + b + c)}$

6. In any $\triangle ABC,$ if $\tan\theta = \frac{2\sqrt{ab}}{a - b}\sin\frac{C}{2},$ prove that $c = (a - b)\sec\theta$

7. In a $\triangle ABC, a=6, b = 3$ and $\cos(A - B) = \frac{4}{5},$ then find its area.

8. In a $\triangle ABC, \angle C=60^\circ$ and $\angle A=75^\circ.$ If $D$ is a point on $AC$ such that area of $\triangle BAD$ is $\sqrt{3}$ times the area of the $\triangle BCD,$ find $\angle ABD$

9. If the sides of a triangle are $3, 5$ and $7,$ prove that the triangle is obtuse angled triangle and find the obtuse angle.

10. In a triangle $ABC,$ if $\angle A = 45^\circ, \angle B = 75^\circ,$ prove that $a + c\sqrt{2} = 2b$

11. In a triangle $ABC, \angle C = 90^\circ, a = 3, b =4$ and $D$ is a point on $AB,$ so that $\angle BCD=30^\circ,$ find the length of $CD.$

12. The sides of a triangle are $4cm, 5cm$ and $6cm.$ Show that the smallest angle is half of the greatest angle.

13. In an isosceles triangle with base $a,$ the vertical angle is $10$ times any of the base angles. Find the length of equal sides of the triangle.

14. The angles of a triangle are in the ratio of $2:3:7,$ then prove that the sides are in the ratio of $\sqrt{2}:2:(\sqrt{3} + 1)$

15. In a triangle $ABC,$ if $\frac{\sin A}{7} = \frac{\sin B}{6} = \frac{\sin C}{5},$ show that $\cos A:\cos B:\cos C = 7:19:25$

16. In any triangle $ABC$ if $\tan\frac{A}{2} = \frac{5}{6}, \tan\frac{B}{2} = \frac{20}{37},$ find $\tan\frac{C}{2}$ and prove that in this triangle $a + c = 2b.$

17. In a triangle $ABC$ if $\angle C=60^\circ,$ prove that $\frac{1}{a + c} + \frac{1}{b + c} = \frac{3}{a + b + c}$

18. If $\alpha, \beta, \gamma$ be the lengths of the altitudes of a triangle $ABC,$ prove that $\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} = \frac{\cot A + \cot B + \cot C}{\Delta},$ where $\Delta$ is the area of the triangle.

19. In a triangle $ABC,$ if $\frac{a}{b} = 2 + \sqrt{3}$ and $\angle C= 60^\circ,$ show that $\angle A = 105^\circ$ and $\angle B=15^\circ.$

20. If two sides of a triangle and the included angle are given by $a = (1 + \sqrt{3}), b = 2$ and $C=60^\circ,$ find the other two angles and the third side.

21. The sides of a triangle are $x, y$ and $\sqrt{x^2 + xy + y^2}.$ prove that the greatest angle is $120^\circ.$

22. The sides of a triangle are $2x + 3, x^2 + 3x + 3$ and $x^2 + 2x,$ prove that greatest amgle is $120^\circ.$

23. In a triangle $ABC,$ if $3a = b + c,$ prove that $\cot\frac{B}{2}\cot\frac{C}{2} = 2$

24. In a triangle $ABC,$ prove that $a\sin\left(\frac{A}{2} + B\right) = (b + c)\sin\frac{A}{2}$

25. In a triangle $ABC,$ prove that $\frac{\cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2}}{\cot A + \cot B + \cot C} = \frac{(a + b + c)^2}{a^2 + b^2 + c^2}$

26. In a triangle $ABC,$ prove that $\frac{b^2 - c^2}{a^2}\sin2A + \frac{c^2 - a^2}{b^2}\sin2B + \frac{a^2 - b^2}{c^2}\sin2C = 0$

27. In a trianlge $ABC,$ prove that $a^3\cos(B - C) + b^3\cos(C - A) + c^3\cos(A - B) = 3abc$

28. In a triangle $ABC,$ prove that $\frac{\cos^2\frac{B - C}{2}}{(b + c)^2} + \frac{\sin^2\frac{B - C}{2}}{(b - c)^2} = \frac{1}{a^2}$

29. In a triangle $ABC,$ prove that $\frac{a}{\cos B\cos C} + \frac{b}{\cos C\cos A} + \frac{c}{\cos A\cos B} = 2a\tan B\tan C\sec A$

30. In a triangle $ABC,$ prove that $(b - c)\cos\frac{A}{2} = a\sin\frac{B - C}{2}$

31. In a triangle $ABC,$ prove that $\tan\left(\frac{A}{2} + B\right) = \frac{c + b}{c - b}\tan \frac{A}{2}$

32. In a triangle $ABC,$ prove that $\tan\frac{A - B}{2} = \frac{a - b}{a + b}\cot\frac{C}{2}$

33. In a triangle $ABC,$ prove that $(b + c)\cos A + (c + a)\cos B + (a + b)\cos C = a + b + c$

34. In a triangle $ABC,$ prove that $\frac{\cos^2B - \cos^2C}{b + c} + \frac{\cos^2C - \cos^2A}{c + a} + \frac{\cos^2A - \cos^2B}{a + b} = 0$

35. In a triangle $ABC,$ prove that $a^3\sin(B - C) + b^3\sin(C - A) + c^3\sin(A - B) = 0$

36. In a triangle $ABC,$ prove that $(b + c - a)\tan\frac{A}{2} = (c + a - b)\tan\frac{B}{2} = (a + b - c)\tan\frac{C}{2}$

37. In a triangle $ABC,$ prove that $1 - \tan\frac{A}{2}\tan\frac{B}{2} = \frac{2c}{a + b + c}$

38. In a triangle $ABC,$ prove that $\frac{\cos2A}{a^2} - \frac{\cos2B}{b^2} = \frac{1}{a^2} - \frac{1}{b^2}$

39. In a triangle $ABC,$ prove that $a^2(\cos^2B - \cos^2C) + b^2(\cos^2C - \cos^2A) + c^2(\cos^2A - \cos^2B) = 0$

40. In a triangle $ABC,$ prove that $\frac{a^2\sin(B - C)}{\sin B + \sin C} + \frac{b^2\sin(C - A)}{\sin C + \sin A} + \frac{c^2\sin(A - B)}{\sin A + \sin B} = 0$

41. In a triangle $ABC,$ prove that $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}$

42. In a triangle $ABC,$ prove that $\frac{\cos A}{a} + \frac{a}{bc} = \frac{\cos B}{b} + \frac{b}{ca} = \frac{\cos C}{c} + \frac{c}{ab}$

43. In a triangle $ABC,$ prove that $(b^2 - c^2)\cot A + (c^2 - a^2)\cot B + (a^2 - b^2)\cot C = 0$

44. In a triangle $ABC,$ prove that $(b - c)\cot\frac{A}{2} + (c - a)\cot\frac{B}{2} + (a - b)\cot\frac{C}{2} = 0$

45.In a triangle $ABC,$ prove that $(a - b)^2\cos^2\frac{C}{2} + (a + b)^2\sin^2\frac{C}{2} = c^2$

1. In a triangle $ABC,$ prove that $\frac{a- b}{a + b} = \cot\frac{A + B}{2}\tan\frac{A - B}{2}$

2. In a triangle $ABC, D$ is the middle point of $BC.$ If $AD$ is perpendicular to $AC,$ prove that $\cos A\cos C = \frac{2(c^2 - a^2)}{3ac}$

3. If $D$ be the middle point of the side $BC$ of the triangle $ABC$ where area is $\Delta$ and $\angle ADB=\theta,$ prove that $\frac{AC^2 - AB^2}{4\Delta} = \cot\theta$

4. $ABCD$ is a trapezium such that $AB$ and $DC$ are parallel and $BC$ is perpendicular to the. If $\angle ADB = \theta, BC = p, CD=q,$ show that $AB = \frac{(p^2 + q^2)\sin\theta}{p\cos\theta + q\sin\theta}$

5. Let $O$ be a point inside a triangle $ABC$ such that $\angle OAB = \angle OBC = \angle OCA = \theta,$ show that $\cot \theta = \cot A + \cot B + \cot C.$

6. The median $AD$ of a triangle $ABC$ is perpendicular to $AB.$ Prove that $\tan A + 2\tan B = 0.$

7. In a triangle $ABC,$ if $\cot A+ \cot B + \cot C = \sqrt{3}$

8. In a triangle $ABC,$ if $(a^2 + b^2)\sin(A - B) = (a^2 - b^2)\sin(A + B)$

9. In a triangle $ABC,$ if $\theta$ be any angle, show that $b\cos\theta = c\cos(A - \theta) + a\cos(C + \theta)$

10. In a triangle $ABC, AD$ is the median. If $\angle BAD = \theta,$ prove that $\cos\theta = 2\cot A + \cot B$

11. The bisector of angle $A$ of a triangle $ABC$ meets $BC$ in $D,$ show that $AD = \frac{2bc}{b + c}\cos \frac{A}{2}$

12. Let $A$ and $B$ be two points on one bank of a straight river and $C$ and $D$ be two points on the other bank, the direction from $A$ to $B$ along the river being the same as from $C$ to $D.$ If $AB = a, \angle CAD = \alpha, \angle DAB = \beta, \angle CBA=\gamma,$ prove that $CD = \frac{a\sin\alpha\sin\gamma}{\sin\beta \sin(\alpha + \beta + \gamma)}$

13. In a triangle $ABC,$ if $2\cos A = \frac{\sin B}{\sin C},$ prove that the triangle is isosceles.

14. If the cosines of two angles of a triangle are inversely proportional to the opposite sides, show that the triangle is either isosceles or right angled.

15. In a triangle $ABC,$ if $a\tan A + b\tan B = (a + b)\tan\frac{A + B}{2},$ prove that the triangle is isosceles.

16. In a triangle $ABC,$ if $\frac{\tan A - \tan B}{\tan A + \tan B} = \frac{c - b}{c},$ prove that $A = 60^\circ$

17. In a triangle $ABC,$ if $c^4 - 2(a^2 + b^2)c^2 + a^4 + a^2b^2 + b^4 = 0,$ prove that $C=60^\circ$ or $120^\circ$

18. In a triangle $ABC,$ if $\frac{\cos A + 2\cos C}{\cos A + 2\cos B} = \frac{\sin B}{\sin C},$ prove that the triangle is either isosceles or right angled.

19. If $A, B, C$ are angles of a $\triangle ABC$ and if $\tan\frac{A}{2}, \tan\frac{B}{2}, \tan\frac{C}{2}$ are in A.P., prove that $\cos A, \cos B, \cos C$ are in A.P.

20. In a triangle $ABC,$ if $a\cos^2\frac{C}{2} + c\cos^2\frac{A}{2} = \frac{3b}{2},$ show that $\cot\frac{A}{2}, \cot\frac{B}{2}, \cot\frac{C}{2}$ are in A.P.

21. If $a^2, b^2, c^2$ are in A.P., then prove that $\cot A, \cot B, \cot C$ are in A.P.

22. The angles $A, B$ and $C$ of a triangle $ABC$ are in A.P. If $2b^2 = 3c^2,$ determine the angle $A.$

23. If in a triangle $ABC, \tan\frac{A}{2}, \tan\frac{B}{2}, \tan\frac{C}{2}$ are in H.P., then show that the sides $a, b, c$ are in A.P.

24. In a triangle $ABC,$ if $\frac{\sin A}{\sin C} = \frac{\sin(A - B)}{\sin(B - C)},$ prove that $a^2, b^2, c^2$ are in A.P.

25. In a triangle $ABC, \sin A, \sin B, \sin C$ are in A.P. show that $3\tan\frac{A}{2}\tan\frac{C}{2} = 1.$

26. In a triangle $ABC,$ if $a^2, b^2, c^2$ are in A.P., show that $\tan A, \tan B, \tan C$ are in H.P.

27. In a triangle $ABC,$ if $a^2, b^2, c^2$ are in A.P., show that $\cot A, \cot B, \cot C$ are in A.P.

28. If the angles $A, B, C$ of a triangle $ABC$ be in A.P. and $b:c = \sqrt{3}:\sqrt{2},$ find the angle $A.$

29. The sides of a triangle are in A.P. and the greatest angle exceeds the least angle by $90^\circ.$ Prove that the sides are in the ratio $\sqrt{7} + 1: \sqrt{7}: \sqrt{7} - 1.$

30. If the sides $a, b, c$ of a triangle are in A.P. and if $a$ is the least side, prove that $\cos A = \frac{4c - 3b}{2c}$

31. The two adjacent sides of a cyclic quadrilateral are $2$ and $5$ nad the angle between them is $60^\circ.$ If the third side is $3,$ find the fourth side.

32. Find the angle $A$ of triangle $ABC,$ in which $(a + b + c)(b + c - a) = 3bc$

33. If in a triangle $ABC, \angle A = \frac{\pi}{3}$ and $AD$ is a median, then prove that $4AD^2 = b^2 + bc + c^2$

34. Prove that the median $AD$ and $BE$ of a $\Delta ABC$ intersect at right angle if $a^2 + b^2 = 5c^2$

35. If in a triangle $ABC, \frac{\tan A}{1} = \frac{\tan B}{2} = \frac{\tan C}{3},$ then prove that $6\sqrt{2}a = 3\sqrt{5}b = 2\sqrt{10}c$

36. The sides of a triangle are $x^2 + x + 1, 2x + 1$ and $x^2 - 1,$ prove that the greatest anngle is $120^\circ.$

37. The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

38. For a triangle $ABC$ having area $12$ sq. cm. and base is $6$ cm. The difference of base angles is $60^\circ.$ Show that angle $A$ opposite to the base is given by $8\sin A - 6\cos A = 3.$

39. In any triangle $ABC,$ if $\cos\theta = \frac{a}{b + c}, \cos\phi = \frac{b}{a + c}, \cos\psi = \frac{c}{a + b}$ where $\theta, \phi$ and $\psi$ lie between $0$ and $\pi,$ prove that $\tan^2\frac{\theta}{2} + \tan^2\frac{\phi}{2} + \tan^2\frac{\psi}{2} = 1.$

40. In a triangle $ABC,$ if $\cos A\cos B + \sin A\sin B\sin C = 1,$ show that the sides are in the proportion $1:1:\sqrt{2}.$

41. The product of the sines of the angles of a triangle is $p$ and the product of their cosines is $q.$ Show that the tangents of the angles are the roots of the equation $qx^3 - px^2 + (1 + q)x - p = 0$

42. In a $\triangle ANC,$ if $\sin^3\theta = \sin(A - \theta)\sin(B - \theta)\sin(C - \theta),$ prove that $\cot\theta = \cot A + \cot B + \cot C.$

43. In a triangle of base $a,$ the ratio of the other two sides is $r(< 1),$ show that the altitude of the triangle is less than or equal to $\frac{ar}{1 - r^2}$

44. Given the base $a$ of a triangle, the opposite angle $A,$ and the product $k^2$ of the other two sides. Solve the triangle and show that there is such triangle if $a < 2k\sin\frac{A}{2}, k$ being positive.

45. A ring $10$ cm in diameter, is suspended from a point $12$ cm above its center by $6$ equal strings, attached at equal intervals. Find the cosine of the angle between consecutive strings.

46. If $2b = 3a$ and $\tan^2\frac{A}{2} = \frac{3}{5},$ prove that there are two values of third side, one of which is double the other.

47. The angles of a triangle are in the ratio $1:2:7,$ prove that the ratio of the greater side to the least side is $\sqrt{5} + 1:\sqrt{5} - 1.$

48. If $f, g, h$ are internal bisectors of the angles of a triangle $ABC,$ show that $\frac{1}{f}\cos\frac{A}{2} + \frac{1}{g}\cos\frac{B}{2} + \frac{1}{h}\cos\frac{C}{2} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$

49. If in a triangle $ABC, BC = 5, CA = 4, AB = 3$ and $D$ and $E$ are points on $BC$ scuh that $BD = DE = EC.$ If $\angle CAB=\theta,$ then prove that $\tan\theta = \frac{3}{8}.$

50. In a triangle $ABC,$ median $AD$ and $CE$ are drawn. If $AD = 5, \angle DAC = \frac{\pi}{8}$ and $\angle ACE = \frac{\pi}{4},$ find the area of the triangle $ABC.$

51. The sides of a triangle are $7, 4\sqrt{3}$ and $\sqrt{13}$ cm. Then prove that the smallest angle is $30^\circ.$

52. In an isosceles, right angled triangle a straight line is drawn from the middle point of one of the equal sides to the opposite angle. Show that it divides the angle in two parts whose cotangents are $2$ and $3.$

53. The sides of a triangle are such that $\frac{a}{1 + m^2n^2} = \frac{b}{m^2 + n^2} = \frac{c}{(1- m^2)(1 + n^2)},$ prove that $A = 2\tan^{-1}\frac{m}{n}, B = 2\tan^{-1}mn$ and $\Delta = \frac{mnbc}{m^2 + n^2}.$

54. The sides $a, b, c$ if a triangle $ABC$ are the roots of the equation $x^3 - px^2 + qx - r = 0,$ prove that its area is $\frac{1}{4}\sqrt{p(4pq - p^3 - 8r)}$

55. Two sides of a triangle are of lengths $\sqrt{6}$ cm and $4$ cm and the angle opposite to the smaller side is $30^\circ.$ How many such triangles are possible? Fine the length of their third side and area.

56. The base of a triangle is divided into three equal parts. If $t_1, t_2, t_3$ be the tangents of the angles subtended by these parts at the opposite vertex, prove that $\left(\frac{1}{t_1} + \frac{1}{t_2}\right)\left(\frac{1}{t_2} + \frac{1}{t_3}\right) = 4\left(1 + \frac{1}{t_2^2}\right)$

57. The three medians of a triangle $ABC$ make angles $\alpha, \beta, \gamma$ with each other, prove that $\cot\alpha + \cot\beta + \cot\gamma + \cot A + \cot B + \cot C = 0.$

58. Perpendiculars are drawn from the angles $A, B, C$ of an acute angled triangle on the opposite sides and produced to meet the circumscribing circle. If these produced parts be $\alpha, \beta, \gamma$ respectively, show that $\frac{a}{\alpha} + \frac{b}{\beta} + \frac{c}{\gamma} = 2(\tan A + \tan B + \tan C)$

59. In a triangle $ABC,$ the vertices $A, B, C$ are at distance $p, q, r$ from the orthocenter respectively. Show that $aqr + brp + cpq = abc$

60. The area of a circular plot of land in the form of a unit circle is to be divided into two equal parts by the arc of a circle whose center is on the circumference of the plot. Show that the radius of the circular arc is given by $\cos\theta$ where $\theta$ is given by $\frac{\pi}{2} = \sin2\theta - 2\theta\cos2\theta$

61. $BC$ is a side of a square, on the perpendicular bisector of $BC,$ two points $P, Q$ are taken, equidistant from the center of square. $BP$ and $CQ$ are joined and cut in $A.$ Prove that in the trangle $ABC,$ $\tan A(\tan B - \tan C)^2 + 8 = 0$

62. If the bisector of the angle $C$ of a triangle $ABC$ cuts $AB$ in $D$ and the circum-circle in $E,$ prove that $CE:DE = (a + b)^2:c^2.$

63. The internal bisectors of the angles of a triangle $ABC$ meet the sides at $D, E$ and $F.$ Show that the area of the triangle $DEF$ is equal to $\frac{2\Delta abc}{(b + c)(c + a)(a + b)}$

64. In a triangle $ABC,$ the measures of the angles $A, B$ and $C$ are $3\alpha, 3\beta$ and $3\gamma$ respectively. $P, Q$ and $R$ are the points within the triangle such that $\angle BAR = \angle RAQ = \angle QAC = \alpha,$ $\angle CBP = \angle PBR = \angle RBA = \beta$ and $\angle ACQ = \angle QCP = \angle PCB = \gamma.$ Show that $AR = 8R\sin\beta\sin\gamma\cos(30^\circ - \gamma)$

65. A circle touches the $x$ axis at $O$ (origin) and intersects the $y$ axis above origin at $B. A$ is a point on that part of cirlce which lies to the right of $OB,$ and the tangents at $A$ and $B$ meet at $T.$ If $\angle AOB = \theta,$ find the angles which the directed line $OA, AT$ and $OB$ makes with $OX.$ If lengths of these lines are $c, t$ and $d$ respectively, show that $c\sin\theta - t(1 + \cos2\theta) = 0$ and $c\cos\theta + t\sin2\theta = d.$

66. If in a triangle $ABC,$ the median $AD$ and the perpendicular $AE$ from the vertex $A$ to the side $BC$ divides the angle $A$ into three equal parts, show that $\cos\frac{A}{3}.\sin^2\frac{A}{3} = \frac{3a^2}{32bc}$

67. In a triangle $ABC,$ if $\cos A + \cos B + \cos C = \frac{3}{2},$ prove that the triangle is equilateral.

68. Prove that a triangle $ABC$ is equilateral if and only if $\tan A + \tan B + \tan C = 3\sqrt{3}.$

69. In a triangle $ABC,$ prove that $(a + b + c)\tan\frac{C}{2} = a\cot\frac{A}{2} + b\cot\frac{B}{2} - c\cot\frac{C}{2}$

70. In a triangle $ABC,$ prove that $\sin^4A + \sin^4B + \sin^4C = \frac{3}{2} + 2\cos A\cos B\cos C + \frac{1}{2}\cos 2A + \cos 2B + \cos 2C$

71. In a triangle $ABC$ prove that $\cos^4A + \cos^4B + \cos^4C = \frac{1}{2} - 2\cos A\cos B\cos C + \frac{1}{2}\cos 2A\cos 2B\cos 2C$

72. In a triangle $ABC,$ prove that $\cot B + \frac{\cos C}{\cos A\sin B} = \cot C + \frac{\cos B}{\cos A\sin C}$

73. In a triangle $ABC,$ prove that $\frac{a\sin(B - C)}{b^2 - c^2} = \frac{b\sin(C - A)}{c^2 - a^2} = \frac{c\sin(A - B)}{a^2 - b^2}$

74. In a triangle $ABC,$ prove that $\sin\frac{B - C}{2} = \frac{b - c}{a}\cos \frac{A}{2}$

75. In a triangle $ABC,$ prove that $\sin^3A\cos(B - C) + \sin^3B\cos(C - A) + \sin^3C\cos(A - B) = 3\sin A\sin B\sin C$

76. In a triangle $ABC,$ prove that $\sin^3A + \sin^3B + \sin^3C = 3\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} + \cos\frac{3A}{2}\cos\frac{3B}{2}\cos\frac{3C}{2}$

77. In a triangle $ABC,$ prove that $\sin3A\sin^3(B - C) + \sin3B\sin^3(C - A) + \sin3C\sin^3(A - B) = 0$

78. In a triangle $ABC,$ prove that $\sin3A\cos^3(B - C) + \sin3B\cos^3(C - A) + \sin3C\cos^3(A - B) = \sin 3A\sin 3B\sin 3C$

79. In a triangle $ABC,$ prove that $\left(\cot\frac{A}{2} + \cot\frac{B}{2}\right)\left(a\sin^2\frac{B}{2} + b\sin^2\frac{A}{2}\right) = c\cot\frac{C}{2}$

80. The sides of a triangle $ABC$ are in A.P. If the angles $A$ and $C$ are the greatest and the smallest angles respectively, prove that $4(1 - \cos A)(1 - \cos C) = \cos A + \cos C$

81. In a triangle $ABC,$ if $a, b, c$ are in H.P., prove that $\sin^2\frac{A}{2}, \sin^2\frac{B}{2}, \sin^2\frac{C}{2}$ are also in H.P.

82. If the sides $a, b, c$ of a triangle $ABC$ be in A.P., prove that $\cos A\cot\frac{A}{2}, \cos B\cot\frac{B}{2}, \cos C\cot\frac{C}{2}$ are in A.P.

83. The sides of a triangle are in A.P. and its area is $\frac{3}{5}$ th of an equilateral triangle of the same perimieter. Prove that the sides are in the ratio $3:5:7.$

84. If the tangents of the angles of a triangle are in A.P., prove that the squares of the sides are in the proportion $x^2(x^2 + 9): (3 + x^2)^2:9(1 + x^2),$ where $x$ is the least or the greatest tangent.

85. If the sides of a triangle are in A.P. and if its greatest angle exceeds the least angle by $\alpha,$ show that the sides are in the ratio $1 - x:1:1 + x$ where $x = \sqrt{\frac{1 - \cos\alpha}{7 - \cos\alpha}}$

86. If the sides of triangle $ABC$ are in G.P. with common ratio $r(r>1),$ show that $r<\frac{1}{2}(\sqrt{5} + 1)$ and $A

87. If $p$ and $q$ be the perpendiculars from the vertices $A$ and $B$ on any line passing through the vertex $C$ of the triangle $ABC$ but not passing through the interior of the angle $ABC,$ prove that $a^2p^2 + b^2q^2 - 2abpq\cos C = a^2b^2\sin^2C$

88. $ABC$ is a triangle, $O$ is a point inside the triangle such that $\angle OAB = \angle OBC = \angle OCA = \theta,$ then show that $\cosec^2\theta = \cosec^2A + \cosec^2B + \cosec^2C$

89. If $x, y, z$ be the lengths of perpendiculars from the circumcenter on the sides $BC, CA, AB$ of a triangle $ABC,$ prove that $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = \frac{abc}{4xyz}$

90. In any triangle $ABC$ if $D$ is any point on the base $BC$ such that $BD:DC = m:n$ and if $AD=x,$ prove that $(m + n)^2x^2 = (m + n)(mb^2 + nc^2) - mna^2$

91. In a triangle $ABC,$ if $\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2},$ prove that the triangle is equilateral.

92. In a triangle $ABC,$ if $\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2} = \frac{1}{8},$ prove that the triangle is equilateral.

93. In a triangle $ABC,$ if $\cos A + 2\cos B + \cos C = 2,$ prove that the sides of the triangle are in A.P.

94. The sides $a, b, c$ of a triangle $ABC$ of a triangle are in A.P., then find the value of $\tan\frac{A}{2} + \tan\frac{C}{2}$ in terms of $\cot\frac{B}{2}.$

95. In a triangle $ABC,$ if $\frac{a - b}{b - c}= \frac{s - a}{s - c},$ prove that $r_1, r_2, r_3$ are in A.P.

96. If the sides $a, b, c$ of a triangle $ABC$ are in G.P., then prove that $x, y, z$ are also in G.P., where $x = (b^2 - c^2)\frac{\tan B + \tan C}{\tan B - \tan C}, y = (c^2 - a^2)\frac{\tan C + \tan A}{\tan C - \tan A}, z = (a^2 - b^2)\frac{\tan A + \tan B}{\tan A - \tan B}$

97. The ex-radii $r_1, r_2, r_3$ of a triangle $ABC$ are in H.P. Show that its sides $a, b, c$ are in A.P.

98. In usual notation, $r_1 = r_2 + r_3 + r,$ prove that the triangle is right-angled.

99. If $A, B, C$ are the angles of a triangle, prove that $\cos A + \cos B + \cos C = 1 + \frac{r}{R}$

100. Show that the radii of the three escribed circles of a triangle are the roots of the equation $x^3 - x^2(4R + r) + xs^2 - rs^2 = 0$

101. The radii $r_1, r_2, r_3$ of escribed circle of a triangle $ABC$ are in H.P. If its area if $24$ sq. cm. and its perimeter is $24$ cm., find the length of its sides.

102. In a triangle $ABC, 8R^2 = a^2 + b^2 + c^2,$ prove that the triangle is right-angled.

103. The radius of the circle passing through the center of the inscribed circle and through the point of the base $BC$ is $\frac{a}{2}\sec\frac{A}{2}$

104. Three circles touch each other externally. The tangents at their point of connect meet at a point whose distance from the point of contact is $4.$ Find the ratio of the product of radii to the sum of of radii of all the circles.

105. In a triangle $ABC,$ if $O$ be the circumcenter and $H,$ the orthocenter, show that $OH = R\sqrt{1 - 8\cos A\cos B\cos C}$

106. Let $ABC$ be a triangle having $O$ and $I$ as its circumcenter an in-center respectively. If $R$ and $r$ be the circumradius and in-radius respectively, then prove that $(IO)^2 = R^2 - 2Rr.$ Further show that the triangle $BIO$ is a right angled triangle if and only if $b$ is the arithmetic means of $a$ and $c.$

107. In any triangle $ABC,$ prove that $\cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2} = \cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2}$

108. Let $ABC$ be a triangle with in-center $I$ and in-radius $r.$ Let $D, E$ and $F$ be the feet of perpendiculars from $I$ to the sides $BC, CA$ and $AB$ respectively. If $r_1, r_2$ and $r_3$ are the radii of circles inscribed in the quadrilaterals $AFIE, BDIF$ and $CEID$ respectively, prove that $\frac{r_1}{r - r_1} + \frac{r_2}{r - r_2} + \frac{r_3}{r - r_3} = \frac{r_1r_2r_3}{(r - r_1)(r - r_2)(r - r_3)}$

109. Show that the line joining the orthocenter to the circumference of a triangle $ABC$ is inclined to $BC$ at an angle $\tan^{-1}\left(\frac{3 - \tan B\tan C}{\tan B - \tan C}\right)$

110. If a circle be drawn touching the inscribed and circumscribed circle of a triangle and $BC$ externally, prove that its radius is $\frac{\Delta}{a}\tan^2\frac{A}{2}.$

111. The bisectors of the angles of a triangle $ABC$ meet its circumcenter in the position $D, E, F.$ Show that the area of the triangle $DEF$ is to that of $ABC$ is $R:2r.$

112. If the bisectors of the angles of a triangle $ABC$ meet the opposite sides in $A', B', C',$ prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}:\cos\frac{A - B}{2}\cos\frac{B - C}{2}\cos\frac{C - A}{2}.$

113. If $a, b, c$ are the sides of a triangle $\lambda a, \lambda b, \lambda c$ the sides of a similar triangle inscribed in the former and $\theta$ the angle between the sides of $a$ and $\lambda a,$ prove that $2\lambda\cos\theta = 1.$

114. If $r$ be the radius of in-circle and $r_1, r_2, r_3$ be the ex-radii of a triangle $ABC,$ prove that $r_1 + r_2 + r_3 - r = 4R$

115. If $r$ be the radius of in-circle and $r_1, r_2, r_3$ be the ex-radii of a triangle $ABC,$ prove that $\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r}$

116. If $r$ be the radius of in-circle and $r_1, r_2, r_3$ be the ex-radii of a triangle $ABC,$ prove that $\frac{1}{r_1^2} + \frac{1}{r_2^2} + \frac{1}{r_3^2} + \frac{1}{r^2} = \frac{a^2 + b^2 + c^2}{\Delta^2}$ where $\Delta$ denotes the area of the triangle $ABC.$

117. If $r$ is the radius of in-circle of a triangle $ABC,$ prove that $r = (s - a)\tan\frac{A}{2} = (s - b)\tan\frac{B}{2} = (s - c)\tan\frac{C}{2}.$

118. If $A, A_1, A_2$ and $A_3$ be respectively the areas of the inscribed and escribed circles of a triangle, prove that $\frac{1}{\sqrt{A}} = \frac{1}{\sqrt{A_1}} + \frac{1}{\sqrt{A_2}} + \frac{1}{\sqrt{A_3}}$

119. In a triangle $ABC,$ prove that $\frac{r_1}{bc} + \frac{r_2}{ca} + \frac{r_3}{ab} = \frac{1}{r} - \frac{1}{2R}.$

120. $ABC$ is an isosceles triangle inscribed in a circle of radius $r.$ If $AB = AC$ and $h$ is the altitude from $A$ to $BC$ then the triangle $ABC$ has perimeter $P = 2(\sqrt{2rh - h^2} + \sqrt{2rh}).$ Find its area.

121. If $p_1, p_2, p_3$ are the altitudes of the triangle $ABC$ from the vertices $A, B, C$ respectively, prove that $\frac{\cos A}{p_1} + \frac{\cos B}{p_2} + \frac{\cos C}{p_3} = \frac{1}{R}.$

122. Three circles whose radii are $a, b, c$ touch one another externally and the tangents at their point of contact meet in a point. Prove that the distance of this point from either of their points of contact is $\sqrt{\frac{abc}{a + b + c}}$

123. In a triangle $ABC,$ prove that $r_1r_2r_3 = r^3\cot^2\frac{A}{2}\cot^2\frac{B}{2}\cot^2\frac{C}{2}.$