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 $(a - b)^2\cos^2\frac{C}{2} + (a + b)^2\sin^2\frac{C}{2} = c^2$

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

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

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

49. $ABCD$ 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 $AB = \frac{(p^2 + q^2)\sin\theta}{p\cos\theta + q\sin\theta}$

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

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

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

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

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

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

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

57. 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 $CD = \frac{a\sin\alpha\sin\gamma}{\sin\beta \sin(\alpha + \beta + \gamma)}$

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

59. 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.

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

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

62. 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 $120^\circ$

63. 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.

64. 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.

65. 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.

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

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

68. 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.

69. 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.

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

71. 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.

72. 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.

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

74. 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 $\sqrt{7} + 1: \sqrt{7}: \sqrt{7} - 1.$

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

76. 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.

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

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

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

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

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

82. 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.

83. 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 $8\sin A - 6\cos A = 3.$

84. 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 $\tan^2\frac{\theta}{2} + \tan^2\frac{\phi}{2} + \tan^2\frac{\psi}{2} = 1.$

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

86. 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 $qx^3 - px^2 + (1 + q)x - p = 0$

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

88. 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 $\frac{ar}{1 - r^2}$

89. 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.

90. 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.

91. 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.

92. 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 $\sqrt{5} + 1:\sqrt{5} - 1.$

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

94. 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 $\tan\theta = \frac{3}{8}.$

95. 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 $ABC.$

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

97. 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 $3.$

98. 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 $\Delta = \frac{mnbc}{m^2 + n^2}.$

99. 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 $\frac{1}{4}\sqrt{p(4pq - p^3 - 8r)}$

100. 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.

101. 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 $\left(\frac{1}{t_1} + \frac{1}{t_2}\right)\left(\frac{1}{t_2} + \frac{1}{t_3}\right) = 4\left(1 + \frac{1}{t_2^2}\right)$

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

103. 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 $\frac{a}{\alpha} + \frac{b}{\beta} + \frac{c}{\gamma} = 2(\tan A + \tan B + \tan C)$

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

105. 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 $\frac{\pi}{2} = \sin2\theta - 2\theta\cos2\theta$

106. $BC$ 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 $ABC,$ $\tan A(\tan B - \tan C)^2 + 8 = 0$

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

108. 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 $\frac{2\Delta abc}{(b + c)(c + a)(a + b)}$

109. 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 $AR = 8R\sin\beta\sin\gamma\cos(30^\circ - \gamma)$

110. 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 $c\cos\theta + t\sin2\theta = d.$

111. 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 $\cos\frac{A}{3}.\sin^2\frac{A}{3} = \frac{3a^2}{32bc}$

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

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

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

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

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

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

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

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

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

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

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

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

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

125. 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 $4(1 - \cos A)(1 - \cos C) = \cos A + \cos C$

126. 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.

127. 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.

128. 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 $3:5:7.$

129. 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.

130. 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 $x = \sqrt{\frac{1 - \cos\alpha}{7 - \cos\alpha}}$

131. 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 $A<B<\frac{\pi}{3}<C$

132. 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 $a^2p^2 + b^2q^2 - 2abpq\cos C = a^2b^2\sin^2C$

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

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

135. 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 $(m + n)^2x^2 = (m + n)(mb^2 + nc^2) - mna^2$

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

137. 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.

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

139. 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 $\cot\frac{B}{2}.$

140. 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.

141. 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 $x = (b^2 - c^2)\frac{\tan B + \tan C}{\tan B - \tan C}, y = (c^2 - a^2)\frac{\tan C + \tan A}{\tan C - \tan A}, z = (a^2 - b^2)\frac{\tan A + \tan B}{\tan A - \tan B}$

142. 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.

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

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

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

146. 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.

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

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

149. 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.

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

151. 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 $c.$

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

153. 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 $\frac{r_1}{r - r_1} + \frac{r_2}{r - r_2} + \frac{r_3}{r - r_3} = \frac{r_1r_2r_3}{(r - r_1)(r - r_2)(r - r_3)}$

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

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

156. 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 $R:2r.$

157. 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 $2\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}:\cos\frac{A - B}{2}\cos\frac{B - C}{2}\cos\frac{C - A}{2}.$

158. 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 $2\lambda\cos\theta = 1.$

159. 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 $r_1 + r_2 + r_3 - r = 4R$

160. 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 $\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r}$

161. 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 $ABC.$

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

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

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

165. $ABC$ 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.

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

167. 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 $\sqrt{\frac{abc}{a + b + c}}$

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

169. In a triangle $ABC,$ prove that $a(rr_1 + r2r_3) = b(rr_2 + r_3r_1) = c(rr_3 + r_1r_2) = abc.$

170. In a triangle $ABC,$ prove that $(r_1 + r_2)\tan\frac{C}{2} = (r_3 - r)\cot\frac{C}{2} = c.$

171. In a triangle $ABC,$ prove that $4R\sin A\sin B\sin C = a\cos A + b\cos B + c\cos C.$

172. In a triangle $ABC,$ prove that $(r_1 - r)(r_2 - r)(r_3 - r) = 4Rr^2$

173. In a triangle $ABC,$ prove that $r^2 + r_1^2 + r_2^2 + r_3^2 = 16R^2 - a^2 - b^2 - c^2$

174. In a triangle $ABC,$ prove that $IA. IB. IC = abc\tan\frac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}$

175. In a triangle $ABC,$ prove that $AI_1 = r_1\cosec\frac{A}{2}$

176. In a triangle $ABC,$ prove that $II_1 = a\sec\frac{A}{2}$

177. In a triangle $ABC,$ prove that $I_2I_3 = a\cosec\frac{A}{2}$

178. In a triangle $ABC,$ if $I$ is the in-center and $I_1, I_2$ and $I_3$ are the centers of the escribed circles, then prove that $II_1.II_2.II_3 = 16R^2r$

179. In a triangle $ABC,$ if $I$ is the in-center and $I_1, I_2$ and $I_3$ are the centers of the escribed circles, then prove that $II_1^2.I_2I_3^2 = II_2^2 + I_3I_1^2 = II_3^2 + I_1I_2^2 = 16R^2$

180. In a triangle $ABC,$ if $O$ is the circumcenter and $I,$ the in-center then prove that $OI^2 = R^2(3 - 2\cos A - 2\cos B - 2\cos C).$

181. In a triangle $ABC,$ if $H$ is the orthocenter and $I$ the in-center then prove that $IH^2 = 2r^2 - 4R^2\cos A\cos B\cos C.$

182. In a triangle $ABC,$ if $O$ is the circumcenter, $G,$ the cetroid and $H,$ the orthocenter then prove that $OG^2 = R^2 - \frac{1}{9}(a^2 + b^2 + c^2).$

183. Given an isosceles triangle with lateral side of length $b,$ base angle $\alpha < \frac{\pi}{4};R, r$ the radii and $O, I$ the centers of the circumcircle and in-circle respectively, then prove that $R = \frac{1}{2}b\cosec\frac{\alpha}{2}.$

184. Given an isosceles triangle with lateral side of length $b,$ base angle $\alpha < \frac{\pi}{4};R, r$ the radii and $O, I$ the centers of the circumcircle and in-circle respectively, then prove that $r = \frac{b\sin2\alpha}{2(1 + \cos\alpha)}$

185. Given an isosceles triangle with lateral side of length $b,$ base angle $\alpha < \frac{\pi}{4};R, r$ the radii and $O, I$ the centers of the circumcircle and in-circle respectively, then prove that $OI = \left|\frac{b\cos\frac{3\alpha}{2}}{2\sin \alpha\cos\frac{\alpha}{2}}\right|$

186. In a triangle $ABC,$ prove that $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \frac{1}{2Rr}$

187. In a triangle $ABC,$ prove that $\frac{r_1}{(s - b)(s - c)} + \frac{r_2}{(s - c)(s - a)} + \frac{r_3}{(s - a)(s - b)} = \frac{3}{r}.$

188. If $\alpha, \beta, \gamma$ are the distances of the vertices of a triangle from the corresponding points of contact with the in-circle, prove that $r^2 = \frac{\alpha\beta\gamma}{\alpha + \beta + \gamma}$

189. Tangents are drawn to the in-circle of triangle $ABC$ which are parallel to its sides. If $x, y, z$ be the lengths of the tangents and $a, b, c$ be the sides of triangle then prove that $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

190. If $t_1, t_2, t_3$ be the length of tangents from the centers of escribed circles to the circumcircle, prove that $\frac{1}{t_1^2} + \frac{1}{t_2^2} + \frac{1}{t_3^2} = \frac{2s}{abc}.$

191. If in a triangle $ABC, \left(1 - \frac{r_1}{r_2}\right)\left(1 - \frac{r_1}{r_3}\right) = 2,$ prove that the triangle is right angled.

192. In a triangle $ABC,$ prove that the area of the in-circle is to the area of the triangle itself is $\pi: \cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2}$

193. Let $A_1, A_2, A_3, \ldots, A_n$ be the vertices of polygon having an $n$ sides such that $\frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4}$ then find the value of $n.$

194. Prove that the sum of radii of the circles, which are respectively inscribed in and circumscibed about a regular polygon of $n$ sides, is $\frac{a}{2}\cot\frac{\pi}{2n},$ where $a$ is the side of the polygon.

195. The sides of a quadrilateral are $3, 4, 5$ and $6$ cms. The sum of a pair of opposite angles is $120^\circ.$ Show that the area of the quadrilateral is $3\sqrt{30}$ sq. cm.

196. The two adjacent sides of a quadrilateral are $2$ and $5$ and the angle between them is $60^\circ.$ If the area of the quadrilateral is $4\sqrt{3},$ find the two remaining sides.

197. A cyclic quadrilateral $ABCD$ of area $\frac{3\sqrt{3}}{4}$ is inscribed in a unit circle. If one of its sides $AB = 1$ and the diagonal $BD = \sqrt{3},$ find lengths of the other sides.

198. If $ABCD$ be a quadrilateral inscribed in a circle, prove that $\tan\frac{B}{2} = \sqrt{\frac{(S - a)(S - b)}{(S - c)(S - d)}}.$

199. $a, b, c$ and $d$ are the sides of a quadrilateral taken in order and $\alpha$ is the angle between diagonals opposite to $b$ or $d,$ prove that the area of the quadrilateral is $\frac{1}{2}(a^2 - b^2 + c^2 - d^2)\tan\alpha$

200. If a quadrilateral can be inscribed in one circle and circumscribed about another circle, prove that its area is $\sqrt{abcd}$ and the radius of the latter circle is $\frac{2\sqrt{abcd}}{a + b + c + d}.$

201. The sides of a quadrilateral which can be inscribed in a circle are $3, 3, 4$ and $4$ cm; find the radii of in-circle and circumcircle.

202. A square whose sides are $2$ cm., has its corners cut away so as to form a regular octagon; find its area.

203. If an equilateral triangle and a regular hexagon have the same perimeter, prove that ratio of their areas is $2:3.$

204. Given that the area of a polygon of $n$ sides circumscribed about a circle is to the area of the circumscribed polygon of $2n$ sides as $3:2,$ find $n.$

205. The area of a polygon of $n$ sides inscribed in a circle is to that of the same number of sides circumscribing the same circle as $3:4.$ Fine the value of $n.$

206. There are two regular polygons, the number of sides in one being the double the number in the other, and an angle of one ploygon is to an angle of the other is $9:8;$ find the number of sides of each polygon.

207. Six equal circles, each of radius $a,$ are placed so that each touches to others, their centers are joined to form a hexagon. Prove that the area which the circles enclose is $2a^2(3\sqrt{3} - \pi).$

208. A cyclic quadrilateral $ABCD$ of area $\frac{3\sqrt{3}}{4}$ is inscribed in a unit circle. If one of its sides $AB = 1$ and the diagonal $BD=\sqrt{3},$ find lengths of the other sides.

209. If $ABCD$ is a cyclic quadrilateral, then prove that $AC.BD = AB.CD + BC.AD$

210. If the number of sides of two regular polygons having the same perimeter be $n$ and $2n$ respectively, prove that their areas are in the ratio $2\cos\frac{\pi}{n}:\left(1 + \cos\frac{\pi}{n}\right).$

211. In a triangle $ABC,$ prove that $\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\leq \frac{1}{8}$

212. The sides of a triangle inscribed in a given circle subtend angles $\alpha, \beta$ and $\gamma$ at the center. Find the minimum value of the arithmetic mean of $\cos\left(\alpha + \frac{\pi}{2}\right), \cos\left(\beta + \frac{\pi}{2}\right)$ and $\cos\left(\gamma + \frac{\pi}{2}\right)$

213. In a triangle $ABC,$ prove that $\tan^2\frac{A}{2}\tan^2\frac{B}{2}\tan^2\frac{C}{2} \geq 1$

214. Let $1<m<3.$ In a triangle $ABC$ if $2b = (m + 1)a$ and $\cos A = \frac{1}{2}\sqrt{\frac{(m - 1)(m + 3)}{m}},$ prove that there are two values of the third side, one of which is $m$ times the other.

215. If $\Delta$ denotes the area of any triangle and $s$ its semiperimeter, prove that $\Delta < \frac{s^2}{4}.$

216. Let $A, B, C$ be three angles such that $A = \frac{\pi}{4}$ and $\tan B\tan C = p.$ Find all possible values of $p$ such that $A, B, C$ are the angles of a triangle.

217. Through the angular points of a triangle straight lines are drawn, which make the same angle $\alpha$ with the opposite side of the triangle. Prove that the area of the triangle formed by them is to the area of the triangle is $4\cos^2\alpha:1$

218. Consider the following statements about a triangle $ABC$

     1. The sides $a,b,c$ and $\Delta$ are rational.

     2. $a, \tan\frac{B}{2}, \tan\frac{ C}{2}$ are rational

     3. $a, \sin A, \sin B, \sin C$ are rational.

     Prove that $1 \Rightarrow 2 \Rightarrow 3 \Rightarrow 1$

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

220. A circle is inscribed in an equilateral triangle of side $a.$ Prove that the area of any square inscribed in this circle is $\frac{a^2}{6}.$

221. In a triangle $ABC, AD$ is the altitude from $A.$ Given $b > c, \angle C = 23^\circ$ and $AD = \frac{abc}{b^2 - c^2},$ then find $\angle B.$

222. In a triangle $ABC, a:b:c = 4:5:6,$ then find the ratio of the radius of the circumcircle to that of in-circle.

223. In a triangle $ABC, \angle B = \frac{\pi}{3}, \angle C= \frac{\pi}{4}$ and $D$ divides $BC$ internally in the ratio of $1:3.$ Prove that $\frac{\sin\angle BAD}{\sin\angle CAD} = \frac{1}{\sqrt{6}}$

224. In a triangle $ABC,$ angle $A$ is greater than angle $B.$ If the measure of angle $A$ and $B$ satisfy the equation $3\sin x - 4\sin^3x - k = 0, 0< k < 1,$ then find the measure of angle $C.$

225. $ABC$ is a triangle such that $\sin(2A + B) = \sin(C - A) = -\sin(B + 2C),$ if $A, B, C$ are in A.P. determine the value of $A, B$ and $C.$

226. In a right angled triangle the hypotenuse is $2\sqrt{2}$ times the length of perpendicular drawn from the opposite vertex on the hypotenuse. Find the two angles.

227. In a triangle $PQR, \angle R = \frac{\pi}{2}.$ If $\tan\frac{P}{2}$ and $\tan\frac{Q}{2}$ are the roots of the equation $ax^2 + bx + c = 0(a\neq 0),$ then prove that $a + b = c.$

228. In a triangle $ABC,$ the medians to the side $BC$ is of length $\frac{1}{\sqrt{1 - 6\sqrt{3}}}$ and it divides the angle $A$ into angles of $30^\circ$ and $45^\circ.$ Find the lngth of side $BC.$

229. If $A, B, C$ are the anngles of an acute-angled triangle, show that $\tan A + \tan B + \tan C \geq 3\sqrt{3}.$

230. In a triangle $ABC, \cos \frac{A}{2} = \frac{1}{2}\sqrt{\frac{b}{c} + \frac{c}{b}},$ show that the square describe on one side of the is equal to twice the rectangle contained by two other sides.

231. If in a triangle $ABC, \theta$ be the angle determined by the relation $\cos\theta = \frac{a - b}{c}.$ Prove that $\cos\frac{A - B}{2} = \frac{(a + b)\sin\theta}{2\sqrt{ab}}$ and $\cos \frac{A+B}{2} = \frac{c\cos\theta}{2\sqrt{ab}}.$

232. If $R$ be the circum-radius and $r$ the in-radius of a triangle $ABC,$ show that $R\geq 2r.$

233. If $\cos A = \tan B, \cos B = \tan C$ and $\cos C = \tan A,$ show that $\sin A = \sin B = \sin C = 2\sin 18^\circ,$ where $A, B, C$ lie between $0$ and $\pi.$

234. In a triangle $ABC,$ prove that $\cot^2A + \cot^2B + \cot^2C\geq 1$

235. In a triangle $ABC,$ prove that $\tan^2A + \tan^2B + \tan^2C \geq 9$

236. In a triangle $ABC,$ prove that $\cosec\frac{A}{2} + \cosec\frac{B}{2} + \cosec\frac{C}{2} \geq 6$

237. In a triangle $ABC,$ prove that $1 < \cos A + \cos B + \cos C \leq \frac{3}{2}$

238. In a triangle $ABC,$ prove that $\cos A\cos B\cos C \leq \frac{1}{8}$

239. Two circles of radii $a$ and $b$ cut each other at an angle $\theta.$ Prove that the length of the common chord is $\frac{2ab\sin\theta}{\sqrt{a^2 + b^22ab\cos\theta}}.$

240. Three equal circles touch one another; find the radius of the circle which touches all the three circles.

241. In a triangle $ABC,$ prove that $\sum_{r=0}^n{}^nC_ra^rb^{n - r}\cos[rB - (n - r)A] = C^n$

242. In a triangle $ABC, \tan A + \tan B + \tan C = k,$ then find the interval in which $k$ should lie so that there exists one isosceles triangle $ABC.$

243. If $\Delta$ be the area and $s,$ the semi-perimeter of a triangle, then prove that $\Delta \leq \frac{s^2}{3\sqrt{3}}.$

244. Show that the tirangle having sides $3x + 4y, 4x + 3y$ and $5x + 5y$ units where $x > 0, y > 0$ is obtuse-angled triangle.

245. Let $ABC$ be a triangle having altitudes $h_1, h_2, h_3$ from the vertices $A, B, C$ respectively and $r$ be the in-radius, then prove that $\frac{h_1 + r}{h_1 - r} + \frac{h_2 + r}{h_2 - r} + \frac{h_3 + r}{h_3 - r}\geq 0.$

246. If $\Delta_0$ be the area of the triangle formed by joining the points of contact of the inscribed circle with the sides of the given triangle, whose area is $\Delta,$ and $\Delta_1, \Delta_2$ and $\Delta_3$ be the corresponding areas for the escribed circles, prove that $\Delta_1 + \Delta_2 + \Delta_3 - \Delta_0 = 2\Delta.$