17. Properties of Triangles’ Problems

  1. The sides of a triangle are 88 cm, 1010 cm and 1212 cm. Prove that the greatest angle is double the smallest angle.

  2. In a ABC,\triangle ABC, if b+c11=c+a12=a+b13,\frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13}, prove that cosA7=cosB19=cosC25\frac{\cos A}{7} = \frac{\cos B}{19} = \frac{\cos C}{25}

  3. If =a2(bc)2,\triangle = a^2 - (b - c)^2, where \triangle is the area of the ABC,\triangle ABC, then prove that tanA=815\tan A = \frac{8}{15}

  4. In a triangle ABC,ABC, the angles A,B,CA, B, C are in A.P. Prove that 2cosAC2=a+ca2ac+c22\cos\frac{A - C}{2} = \frac{a + c}{\sqrt{a^2 - ac + c^2}}

  5. If p1,p2,p3p_1, p_2, p_3 be the altitudes of a triangle ABCABC from the vertices A,B,CA, B, C respectively and Δ\Delta be the area of the triangle, prove that 1p1+1p21p3=2abcos2C2Δ(a+b+c)\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 ABC,\triangle ABC, if tanθ=2ababsinC2,\tan\theta = \frac{2\sqrt{ab}}{a - b}\sin\frac{C}{2}, prove that c=(ab)secθc = (a - b)\sec\theta

  7. In a ABC,a=6,b=3\triangle ABC, a=6, b = 3 and cos(AB)=45,\cos(A - B) = \frac{4}{5}, then find its area.

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

  9. If the sides of a triangle are 3,53, 5 and 7,7, prove that the triangle is obtuse angled triangle and find the obtuse angle.

  10. In a triangle ABC,ABC, if A=45,B=75,\angle A = 45^\circ, \angle B = 75^\circ, prove that a+c2=2ba + c\sqrt{2} = 2b

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

  12. The sides of a triangle are 4cm,5cm4cm, 5cm and 6cm.6cm. Show that the smallest angle is half of the greatest angle.

  13. In an isosceles triangle with base a,a, the vertical angle is 1010 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 2:3:7,2:3:7, then prove that the sides are in the ratio of 2:2:(3+1)\sqrt{2}:2:(\sqrt{3} + 1)

  15. In a triangle ABC,ABC, if sinA7=sinB6=sinC5,\frac{\sin A}{7} = \frac{\sin B}{6} = \frac{\sin C}{5}, show that cosA:cosB:cosC=7:19:25\cos A:\cos B:\cos C = 7:19:25

  16. In any triangle ABCABC if tanA2=56,tanB2=2037,\tan\frac{A}{2} = \frac{5}{6}, \tan\frac{B}{2} = \frac{20}{37}, find tanC2\tan\frac{C}{2} and prove that in this triangle a+c=2b.a + c = 2b.

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

  18. If α,β,γ\alpha, \beta, \gamma be the lengths of the altitudes of a triangle ABC,ABC, prove that 1α2+1β2+1γ2=cotA+cotB+cotCΔ,\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} = \frac{\cot A + \cot B + \cot C}{\Delta}, where Δ\Delta is the area of the triangle.

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

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

  21. The sides of a triangle are x,yx, y and x2+xy+y2.\sqrt{x^2 + xy + y^2}. prove that the greatest angle is 120.120^\circ.

  22. The sides of a triangle are 2x+3,x2+3x+32x + 3, x^2 + 3x + 3 and x2+2x,x^2 + 2x, prove that greatest amgle is 120.120^\circ.

  23. In a triangle ABC,ABC, if 3a=b+c,3a = b + c, prove that cotB2cotC2=2\cot\frac{B}{2}\cot\frac{C}{2} = 2

  24. In a triangle ABC,ABC, prove that asin(A2+B)=(b+c)sinA2a\sin\left(\frac{A}{2} + B\right) = (b + c)\sin\frac{A}{2}

  25. In a triangle ABC,ABC, prove that cotA2+cotB2+cotC2cotA+cotB+cotC=(a+b+c)2a2+b2+c2\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,ABC, prove that b2c2a2sin2A+c2a2b2sin2B+a2b2c2sin2C=0\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,ABC, prove that a3cos(BC)+b3cos(CA)+c3cos(AB)=3abca^3\cos(B - C) + b^3\cos(C - A) + c^3\cos(A - B) = 3abc

  28. In a triangle ABC,ABC, prove that cos2BC2(b+c)2+sin2BC2(bc)2=1a2\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,ABC, prove that acosBcosC+bcosCcosA+ccosAcosB=2atanBtanCsecA\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,ABC, prove that (bc)cosA2=asinBC2(b - c)\cos\frac{A}{2} = a\sin\frac{B - C}{2}

  31. In a triangle ABC,ABC, prove that tan(A2+B)=c+bcbtanA2\tan\left(\frac{A}{2} + B\right) = \frac{c + b}{c - b}\tan \frac{A}{2}

  32. In a triangle ABC,ABC, prove that tanAB2=aba+bcotC2\tan\frac{A - B}{2} = \frac{a - b}{a + b}\cot\frac{C}{2}

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

  34. In a triangle ABC,ABC, prove that cos2Bcos2Cb+c+cos2Ccos2Ac+a+cos2Acos2Ba+b=0\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,ABC, prove that a3sin(BC)+b3sin(CA)+c3sin(AB)=0a^3\sin(B - C) + b^3\sin(C - A) + c^3\sin(A - B) = 0

  36. In a triangle ABC,ABC, prove that (b+ca)tanA2=(c+ab)tanB2=(a+bc)tanC2(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,ABC, prove that 1tanA2tanB2=2ca+b+c1 - \tan\frac{A}{2}\tan\frac{B}{2} = \frac{2c}{a + b + c}

  38. In a triangle ABC,ABC, prove that cos2Aa2cos2Bb2=1a21b2\frac{\cos2A}{a^2} - \frac{\cos2B}{b^2} = \frac{1}{a^2} - \frac{1}{b^2}

  39. In a triangle ABC,ABC, prove that a2(cos2Bcos2C)+b2(cos2Ccos2A)+c2(cos2Acos2B)=0a^2(\cos^2B - \cos^2C) + b^2(\cos^2C - \cos^2A) + c^2(\cos^2A - \cos^2B) = 0

  40. In a triangle ABC,ABC, prove that a2sin(BC)sinB+sinC+b2sin(CA)sinC+sinA+c2sin(AB)sinA+sinB=0\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,ABC, prove that cosAa+cosBb+cosCc=a2+b2+c22abc\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,ABC, prove that cosAa+abc=cosBb+bca=cosCc+cab\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,ABC, prove that (b2c2)cotA+(c2a2)cotB+(a2b2)cotC=0(b^2 - c^2)\cot A + (c^2 - a^2)\cot B + (a^2 - b^2)\cot C = 0

  44. In a triangle ABC,ABC, prove that (bc)cotA2+(ca)cotB2+(ab)cotC2=0(b - c)\cot\frac{A}{2} + (c - a)\cot\frac{B}{2} + (a - b)\cot\frac{C}{2} = 0

45.In a triangle ABC,ABC, prove that (ab)2cos2C2+(a+b)2sin2C2=c2(a - b)^2\cos^2\frac{C}{2} + (a + b)^2\sin^2\frac{C}{2} = c^2

  1. In a triangle ABC,ABC, prove that aba+b=cotA+B2tanAB2\frac{a- b}{a + b} = \cot\frac{A + B}{2}\tan\frac{A - B}{2}

  2. In a triangle ABC,DABC, D is the middle point of BC.BC. If ADAD is perpendicular to AC,AC, prove that cosAcosC=2(c2a2)3ac\cos A\cos C = \frac{2(c^2 - a^2)}{3ac}

  3. If DD be the middle point of the side BCBC of the triangle ABCABC where area is Δ\Delta and ADB=θ,\angle ADB=\theta, prove that AC2AB24Δ=cotθ\frac{AC^2 - AB^2}{4\Delta} = \cot\theta

  4. ABCDABCD is a trapezium such that ABAB and DCDC are parallel and BCBC is perpendicular to the. If ADB=θ,BC=p,CD=q,\angle ADB = \theta, BC = p, CD=q, show that AB=(p2+q2)sinθpcosθ+qsinθAB = \frac{(p^2 + q^2)\sin\theta}{p\cos\theta + q\sin\theta}

  5. Let OO be a point inside a triangle ABCABC such that OAB=OBC=OCA=θ,\angle OAB = \angle OBC = \angle OCA = \theta, show that cotθ=cotA+cotB+cotC.\cot \theta = \cot A + \cot B + \cot C.

  6. The median ADAD of a triangle ABCABC is perpendicular to AB.AB. Prove that tanA+2tanB=0.\tan A + 2\tan B = 0.

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

  8. In a triangle ABC,ABC, if (a2+b2)sin(AB)=(a2b2)sin(A+B)(a^2 + b^2)\sin(A - B) = (a^2 - b^2)\sin(A + B)

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

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

  11. The bisector of angle AA of a triangle ABCABC meets BCBC in D,D, show that AD=2bcb+ccosA2AD = \frac{2bc}{b + c}\cos \frac{A}{2}

  12. Let AA and BB be two points on one bank of a straight river and CC and DD be two points on the other bank, the direction from AA to BB along the river being the same as from CC to D.D. If AB=a,CAD=α,DAB=β,CBA=γ,AB = a, \angle CAD = \alpha, \angle DAB = \beta, \angle CBA=\gamma, prove that CD=asinαsinγsinβsin(α+β+γ)CD = \frac{a\sin\alpha\sin\gamma}{\sin\beta \sin(\alpha + \beta + \gamma)}

  13. In a triangle ABC,ABC, if 2cosA=sinBsinC,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,ABC, if atanA+btanB=(a+b)tanA+B2,a\tan A + b\tan B = (a + b)\tan\frac{A + B}{2}, prove that the triangle is isosceles.

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

  17. In a triangle ABC,ABC, if c42(a2+b2)c2+a4+a2b2+b4=0,c^4 - 2(a^2 + b^2)c^2 + a^4 + a^2b^2 + b^4 = 0, prove that C=60C=60^\circ or 120120^\circ

  18. In a triangle ABC,ABC, if cosA+2cosCcosA+2cosB=sinBsinC,\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,CA, B, C are angles of a ABC\triangle ABC and if tanA2,tanB2,tanC2\tan\frac{A}{2}, \tan\frac{B}{2}, \tan\frac{C}{2} are in A.P., prove that cosA,cosB,cosC\cos A, \cos B, \cos C are in A.P.

  20. In a triangle ABC,ABC, if acos2C2+ccos2A2=3b2,a\cos^2\frac{C}{2} + c\cos^2\frac{A}{2} = \frac{3b}{2}, show that cotA2,cotB2,cotC2\cot\frac{A}{2}, \cot\frac{B}{2}, \cot\frac{C}{2} are in A.P.

  21. If a2,b2,c2a^2, b^2, c^2 are in A.P., then prove that cotA,cotB,cotC\cot A, \cot B, \cot C are in A.P.

  22. The angles A,BA, B and CC of a triangle ABCABC are in A.P. If 2b2=3c2,2b^2 = 3c^2, determine the angle A.A.

  23. If in a triangle ABC,tanA2,tanB2,tanC2ABC, \tan\frac{A}{2}, \tan\frac{B}{2}, \tan\frac{C}{2} are in H.P., then show that the sides a,b,ca, b, c are in A.P.

  24. In a triangle ABC,ABC, if sinAsinC=sin(AB)sin(BC),\frac{\sin A}{\sin C} = \frac{\sin(A - B)}{\sin(B - C)}, prove that a2,b2,c2a^2, b^2, c^2 are in A.P.

  25. In a triangle ABC,sinA,sinB,sinCABC, \sin A, \sin B, \sin C are in A.P. show that 3tanA2tanC2=1.3\tan\frac{A}{2}\tan\frac{C}{2} = 1.

  26. In a triangle ABC,ABC, if a2,b2,c2a^2, b^2, c^2 are in A.P., show that tanA,tanB,tanC\tan A, \tan B, \tan C are in H.P.

  27. In a triangle ABC,ABC, if a2,b2,c2a^2, b^2, c^2 are in A.P., show that cotA,cotB,cotC\cot A, \cot B, \cot C are in A.P.

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

  29. The sides of a triangle are in A.P. and the greatest angle exceeds the least angle by 90.90^\circ. Prove that the sides are in the ratio 7+1:7:71.\sqrt{7} + 1: \sqrt{7}: \sqrt{7} - 1.

  30. If the sides a,b,ca, b, c of a triangle are in A.P. and if aa is the least side, prove that cosA=4c3b2c\cos A = \frac{4c - 3b}{2c}

  31. The two adjacent sides of a cyclic quadrilateral are 22 and 55 nad the angle between them is 60.60^\circ. If the third side is 3,3, find the fourth side.

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

  33. If in a triangle ABC,A=π3ABC, \angle A = \frac{\pi}{3} and ADAD is a median, then prove that 4AD2=b2+bc+c24AD^2 = b^2 + bc + c^2

  34. Prove that the median ADAD and BEBE of a ΔABC\Delta ABC intersect at right angle if a2+b2=5c2a^2 + b^2 = 5c^2

  35. If in a triangle ABC,tanA1=tanB2=tanC3,ABC, \frac{\tan A}{1} = \frac{\tan B}{2} = \frac{\tan C}{3}, then prove that 62a=35b=210c6\sqrt{2}a = 3\sqrt{5}b = 2\sqrt{10}c

  36. The sides of a triangle are x2+x+1,2x+1x^2 + x + 1, 2x + 1 and x21,x^2 - 1, prove that the greatest anngle is 120.120^\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 ABCABC having area 1212 sq. cm. and base is 66 cm. The difference of base angles is 60.60^\circ. Show that angle AA opposite to the base is given by 8sinA6cosA=3.8\sin A - 6\cos A = 3.

  39. In any triangle ABC,ABC, if cosθ=ab+c,cosϕ=ba+c,cosψ=ca+b\cos\theta = \frac{a}{b + c}, \cos\phi = \frac{b}{a + c}, \cos\psi = \frac{c}{a + b} where θ,ϕ\theta, \phi and ψ\psi lie between 00 and π,\pi, prove that tan2θ2+tan2ϕ2+tan2ψ2=1.\tan^2\frac{\theta}{2} + \tan^2\frac{\phi}{2} + \tan^2\frac{\psi}{2} = 1.

  40. In a triangle ABC,ABC, if cosAcosB+sinAsinBsinC=1,\cos A\cos B + \sin A\sin B\sin C = 1, show that the sides are in the proportion 1:1:2.1:1:\sqrt{2}.

  41. The product of the sines of the angles of a triangle is pp and the product of their cosines is q.q. Show that the tangents of the angles are the roots of the equation qx3px2+(1+q)xp=0qx^3 - px^2 + (1 + q)x - p = 0

  42. In a ANC,\triangle ANC, if sin3θ=sin(Aθ)sin(Bθ)sin(Cθ),\sin^3\theta = \sin(A - \theta)\sin(B - \theta)\sin(C - \theta), prove that cotθ=cotA+cotB+cotC.\cot\theta = \cot A + \cot B + \cot C.

  43. In a triangle of base a,a, the ratio of the other two sides is r(<1),r(< 1), show that the altitude of the triangle is less than or equal to ar1r2\frac{ar}{1 - r^2}

  44. Given the base aa of a triangle, the opposite angle A,A, and the product k2k^2 of the other two sides. Solve the triangle and show that there is such triangle if a<2ksinA2,ka < 2k\sin\frac{A}{2}, k being positive.

  45. A ring 1010 cm in diameter, is suspended from a point 1212 cm above its center by 66 equal strings, attached at equal intervals. Find the cosine of the angle between consecutive strings.

  46. If 2b=3a2b = 3a and tan2A2=35,\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 1:2:7,1:2:7, prove that the ratio of the greater side to the least side is 5+1:51.\sqrt{5} + 1:\sqrt{5} - 1.

  48. If f,g,hf, g, h are internal bisectors of the angles of a triangle ABC,ABC, show that 1fcosA2+1gcosB2+1hcosC2=1a+1b+1c.\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=3ABC, BC = 5, CA = 4, AB = 3 and DD and EE are points on BCBC scuh that BD=DE=EC.BD = DE = EC. If CAB=θ,\angle CAB=\theta, then prove that tanθ=38.\tan\theta = \frac{3}{8}.

  50. In a triangle ABC,ABC, median ADAD and CECE are drawn. If AD=5,DAC=π8AD = 5, \angle DAC = \frac{\pi}{8} and ACE=π4,\angle ACE = \frac{\pi}{4}, find the area of the triangle ABC.ABC.

  51. The sides of a triangle are 7,437, 4\sqrt{3} and 13\sqrt{13} cm. Then prove that the smallest angle is 30.30^\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 22 and 3.3.

  53. The sides of a triangle are such that a1+m2n2=bm2+n2=c(1m2)(1+n2),\frac{a}{1 + m^2n^2} = \frac{b}{m^2 + n^2} = \frac{c}{(1- m^2)(1 + n^2)}, prove that A=2tan1mn,B=2tan1mnA = 2\tan^{-1}\frac{m}{n}, B = 2\tan^{-1}mn and Δ=mnbcm2+n2.\Delta = \frac{mnbc}{m^2 + n^2}.

  54. The sides a,b,ca, b, c if a triangle ABCABC are the roots of the equation x3px2+qxr=0,x^3 - px^2 + qx - r = 0, prove that its area is 14p(4pqp38r)\frac{1}{4}\sqrt{p(4pq - p^3 - 8r)}

  55. Two sides of a triangle are of lengths 6\sqrt{6} cm and 44 cm and the angle opposite to the smaller side is 30.30^\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 t1,t2,t3t_1, t_2, t_3 be the tangents of the angles subtended by these parts at the opposite vertex, prove that (1t1+1t2)(1t2+1t3)=4(1+1t22)\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 ABCABC make angles α,β,γ\alpha, \beta, \gamma with each other, prove that cotα+cotβ+cotγ+cotA+cotB+cotC=0.\cot\alpha + \cot\beta + \cot\gamma + \cot A + \cot B + \cot C = 0.

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

  59. In a triangle ABC,ABC, the vertices A,B,CA, B, C are at distance p,q,rp, q, r from the orthocenter respectively. Show that aqr+brp+cpq=abcaqr + 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 cosθ\cos\theta where θ\theta is given by π2=sin2θ2θcos2θ\frac{\pi}{2} = \sin2\theta - 2\theta\cos2\theta

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

  62. If the bisector of the angle CC of a triangle ABCABC cuts ABAB in DD and the circum-circle in E,E, prove that CE:DE=(a+b)2:c2.CE:DE = (a + b)^2:c^2.

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

  64. In a triangle ABC,ABC, the measures of the angles A,BA, B and CC are 3α,3β3\alpha, 3\beta and 3γ3\gamma respectively. P,QP, Q and RR are the points within the triangle such that BAR=RAQ=QAC=α,\angle BAR = \angle RAQ = \angle QAC = \alpha, CBP=PBR=RBA=β\angle CBP = \angle PBR = \angle RBA = \beta and ACQ=QCP=PCB=γ.\angle ACQ = \angle QCP = \angle PCB = \gamma. Show that AR=8Rsinβsinγcos(30γ)AR = 8R\sin\beta\sin\gamma\cos(30^\circ - \gamma)

  65. A circle touches the xx axis at OO (origin) and intersects the yy axis above origin at B.AB. A is a point on that part of cirlce which lies to the right of OB,OB, and the tangents at AA and BB meet at T.T. If AOB=θ,\angle AOB = \theta, find the angles which the directed line OA,ATOA, AT and OBOB makes with OX.OX. If lengths of these lines are c,tc, t and dd respectively, show that csinθt(1+cos2θ)=0c\sin\theta - t(1 + \cos2\theta) = 0 and ccosθ+tsin2θ=d.c\cos\theta + t\sin2\theta = d.

  66. If in a triangle ABC,ABC, the median ADAD and the perpendicular AEAE from the vertex AA to the side BCBC divides the angle AA into three equal parts, show that cosA3.sin2A3=3a232bc\cos\frac{A}{3}.\sin^2\frac{A}{3} = \frac{3a^2}{32bc}

  67. In a triangle ABC,ABC, if cosA+cosB+cosC=32,\cos A + \cos B + \cos C = \frac{3}{2}, prove that the triangle is equilateral.

  68. Prove that a triangle ABCABC is equilateral if and only if tanA+tanB+tanC=33.\tan A + \tan B + \tan C = 3\sqrt{3}.

  69. In a triangle ABC,ABC, prove that (a+b+c)tanC2=acotA2+bcotB2ccotC2(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,ABC, prove that sin4A+sin4B+sin4C=32+2cosAcosBcosC+12cos2A+cos2B+cos2C\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 ABCABC prove that cos4A+cos4B+cos4C=122cosAcosBcosC+12cos2Acos2Bcos2C\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,ABC, prove that cotB+cosCcosAsinB=cotC+cosBcosAsinC\cot B + \frac{\cos C}{\cos A\sin B} = \cot C + \frac{\cos B}{\cos A\sin C}

  73. In a triangle ABC,ABC, prove that asin(BC)b2c2=bsin(CA)c2a2=csin(AB)a2b2\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,ABC, prove that sinBC2=bcacosA2\sin\frac{B - C}{2} = \frac{b - c}{a}\cos \frac{A}{2}

  75. In a triangle ABC,ABC, prove that sin3Acos(BC)+sin3Bcos(CA)+sin3Ccos(AB)=3sinAsinBsinC\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,ABC, prove that sin3A+sin3B+sin3C=3cosA2cosB2cosC2+cos3A2cos3B2cos3C2\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,ABC, prove that sin3Asin3(BC)+sin3Bsin3(CA)+sin3Csin3(AB)=0\sin3A\sin^3(B - C) + \sin3B\sin^3(C - A) + \sin3C\sin^3(A - B) = 0

  78. In a triangle ABC,ABC, prove that sin3Acos3(BC)+sin3Bcos3(CA)+sin3Ccos3(AB)=sin3Asin3Bsin3C\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,ABC, prove that (cotA2+cotB2)(asin2B2+bsin2A2)=ccotC2\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 ABCABC are in A.P. If the angles AA and CC are the greatest and the smallest angles respectively, prove that 4(1cosA)(1cosC)=cosA+cosC4(1 - \cos A)(1 - \cos C) = \cos A + \cos C

  81. In a triangle ABC,ABC, if a,b,ca, b, c are in H.P., prove that sin2A2,sin2B2,sin2C2\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,ca, b, c of a triangle ABCABC be in A.P., prove that cosAcotA2,cosBcotB2,cosCcotC2\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 35\frac{3}{5} th of an equilateral triangle of the same perimieter. Prove that the sides are in the ratio 3:5:7.3:5:7.

  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 x2(x2+9):(3+x2)2:9(1+x2),x^2(x^2 + 9): (3 + x^2)^2:9(1 + x^2), where xx 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 α,\alpha, show that the sides are in the ratio 1x:1:1+x1 - x:1:1 + x where x=1cosα7cosαx = \sqrt{\frac{1 - \cos\alpha}{7 - \cos\alpha}}

  86. If the sides of triangle ABCABC are in G.P. with common ratio r(r>1),r(r>1), show that r<12(5+1)r<\frac{1}{2}(\sqrt{5} + 1) and A<B<π3<CA<B<\frac{\pi}{3}<C

  87. If pp and qq be the perpendiculars from the vertices AA and BB on any line passing through the vertex CC of the triangle ABCABC but not passing through the interior of the angle ABC,ABC, prove that a2p2+b2q22abpqcosC=a2b2sin2Ca^2p^2 + b^2q^2 - 2abpq\cos C = a^2b^2\sin^2C

  88. ABCABC is a triangle, OO is a point inside the triangle such that OAB=OBC=OCA=θ,\angle OAB = \angle OBC = \angle OCA = \theta, then show that cosec2θ=cosec2A+cosec2B+cosec2C\cosec^2\theta = \cosec^2A + \cosec^2B + \cosec^2C

  89. If x,y,zx, y, z be the lengths of perpendiculars from the circumcenter on the sides BC,CA,ABBC, CA, AB of a triangle ABC,ABC, prove that ax+by+cz=abc4xyz\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = \frac{abc}{4xyz}

  90. In any triangle ABCABC if DD is any point on the base BCBC such that BD:DC=m:nBD:DC = m:n and if AD=x,AD=x, prove that (m+n)2x2=(m+n)(mb2+nc2)mna2(m + n)^2x^2 = (m + n)(mb^2 + nc^2) - mna^2

  91. In a triangle ABC,ABC, if sinA+sinB+sinC=332,\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2}, prove that the triangle is equilateral.

  92. In a triangle ABC,ABC, if sinA2sinB2sinC2=18,\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,ABC, if cosA+2cosB+cosC=2,\cos A + 2\cos B + \cos C = 2, prove that the sides of the triangle are in A.P.

  94. The sides a,b,ca, b, c of a triangle ABCABC of a triangle are in A.P., then find the value of tanA2+tanC2\tan\frac{A}{2} + \tan\frac{C}{2} in terms of cotB2.\cot\frac{B}{2}.

  95. In a triangle ABC,ABC, if abbc=sasc,\frac{a - b}{b - c}= \frac{s - a}{s - c}, prove that r1,r2,r3r_1, r_2, r_3 are in A.P.

  96. If the sides a,b,ca, b, c of a triangle ABCABC are in G.P., then prove that x,y,zx, y, z are also in G.P., where x=(b2c2)tanB+tanCtanBtanC,y=(c2a2)tanC+tanAtanCtanA,z=(a2b2)tanA+tanBtanAtanBx = (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 r1,r2,r3r_1, r_2, r_3 of a triangle ABCABC are in H.P. Show that its sides a,b,ca, b, c are in A.P.

  98. In usual notation, r1=r2+r3+r,r_1 = r_2 + r_3 + r, prove that the triangle is right-angled.

  99. If A,B,CA, B, C are the angles of a triangle, prove that cosA+cosB+cosC=1+rR\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 x3x2(4R+r)+xs2rs2=0x^3 - x^2(4R + r) + xs^2 - rs^2 = 0

  101. The radii r1,r2,r3r_1, r_2, r_3 of escribed circle of a triangle ABCABC are in H.P. If its area if 2424 sq. cm. and its perimeter is 2424 cm., find the length of its sides.

  102. In a triangle ABC,8R2=a2+b2+c2,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 BCBC is a2secA2\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 4.4. Find the ratio of the product of radii to the sum of of radii of all the circles.

  105. In a triangle ABC,ABC, if OO be the circumcenter and H,H, the orthocenter, show that OH=R18cosAcosBcosCOH = R\sqrt{1 - 8\cos A\cos B\cos C}

  106. Let ABCABC be a triangle having OO and II as its circumcenter an in-center respectively. If RR and rr be the circumradius and in-radius respectively, then prove that (IO)2=R22Rr.(IO)^2 = R^2 - 2Rr. Further show that the triangle BIOBIO is a right angled triangle if and only if bb is the arithmetic means of aa and c.c.

  107. In any triangle ABC,ABC, prove that cotA2+cotB2+cotC2=cotA2cotB2cotC2\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 ABCABC be a triangle with in-center II and in-radius r.r. Let D,ED, E and FF be the feet of perpendiculars from II to the sides BC,CABC, CA and ABAB respectively. If r1,r2r_1, r_2 and r3r_3 are the radii of circles inscribed in the quadrilaterals AFIE,BDIFAFIE, BDIF and CEIDCEID respectively, prove that r1rr1+r2rr2+r3rr3=r1r2r3(rr1)(rr2)(rr3)\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 ABCABC is inclined to BCBC at an angle tan1(3tanBtanCtanBtanC)\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 BCBC externally, prove that its radius is Δatan2A2.\frac{\Delta}{a}\tan^2\frac{A}{2}.

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

  112. If the bisectors of the angles of a triangle ABCABC meet the opposite sides in A,B,C,A', B', C', prove that the ratio of the areas of the triangles ABCA'B'C' and ABCABC is 2sinA2sinB2sinC2:cosAB2cosBC2cosCA2.2\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,ca, b, c are the sides of a triangle λa,λb,λc\lambda a, \lambda b, \lambda c the sides of a similar triangle inscribed in the former and θ\theta the angle between the sides of aa and λa,\lambda a, prove that 2λcosθ=1.2\lambda\cos\theta = 1.

  114. If rr be the radius of in-circle and r1,r2,r3r_1, r_2, r_3 be the ex-radii of a triangle ABC,ABC, prove that r1+r2+r3r=4Rr_1 + r_2 + r_3 - r = 4R

  115. If rr be the radius of in-circle and r1,r2,r3r_1, r_2, r_3 be the ex-radii of a triangle ABC,ABC, prove that 1r1+1r2+1r3=1r\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r}

  116. If rr be the radius of in-circle and r1,r2,r3r_1, r_2, r_3 be the ex-radii of a triangle ABC,ABC, prove that 1r12+1r22+1r32+1r2=a2+b2+c2Δ2\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 Δ\Delta denotes the area of the triangle ABC.ABC.

  117. If rr is the radius of in-circle of a triangle ABC,ABC, prove that r=(sa)tanA2=(sb)tanB2=(sc)tanC2.r = (s - a)\tan\frac{A}{2} = (s - b)\tan\frac{B}{2} = (s - c)\tan\frac{C}{2}.

  118. If A,A1,A2A, A_1, A_2 and A3A_3 be respectively the areas of the inscribed and escribed circles of a triangle, prove that 1A=1A1+1A2+1A3\frac{1}{\sqrt{A}} = \frac{1}{\sqrt{A_1}} + \frac{1}{\sqrt{A_2}} + \frac{1}{\sqrt{A_3}}

  119. In a triangle ABC,ABC, prove that r1bc+r2ca+r3ab=1r12R.\frac{r_1}{bc} + \frac{r_2}{ca} + \frac{r_3}{ab} = \frac{1}{r} - \frac{1}{2R}.

  120. ABCABC is an isosceles triangle inscribed in a circle of radius r.r. If AB=ACAB = AC and hh is the altitude from AA to BCBC then the triangle ABCABC has perimeter P=2(2rhh2+2rh).P = 2(\sqrt{2rh - h^2} + \sqrt{2rh}). Find its area.

  121. If p1,p2,p3p_1, p_2, p_3 are the altitudes of the triangle ABCABC from the vertices A,B,CA, B, C respectively, prove that cosAp1+cosBp2+cosCp3=1R.\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,ca, 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 abca+b+c\sqrt{\frac{abc}{a + b + c}}

  123. In a triangle ABC,ABC, prove that r1r2r3=r3cot2A2cot2B2cot2C2.r_1r_2r_3 = r^3\cot^2\frac{A}{2}\cot^2\frac{B}{2}\cot^2\frac{C}{2}.