69. From one angle of a triangle, perpendiculars are dropped on the external bisectors of the other two angles; prove that the distance between the feet of these perpendiculars is equal to half the sum of the sides of the triangle. 61. A, B, P, Q, R, are five points in the circumference of a circle ; p, q, r, are the intersections of perpendiculars of the triangles ABP, ABQ, ABR respectively; prove that the triangles PQR, pqr are similar, equal, and similarly placed. 62. AD, BE, CF are perpendiculars from the angular points of a triangle on the opposite sides, intersecting in P. Prove that the rectangle AP, BC is equal to the sum of the rectangles PE, AC and PF, AB. 63. ABC is a triangle, and AD, AE, are drawn to points D, E, in the base, so as to make equal angles with AB, AC, respectively. Shew that the square on AB is to the square on AC as the rectangle BD, BE is to the rectangle CD, CE. 64. Find a straight line, such that the perpendiculars, let fall upon it from three given points, shall be in a given ratio to each other. 65. Find a fourth proportional to three given similar triangles. 66. If the sides of a triangle be bisected, and the points joined with the opposite angles, the joining lines shall divide each other proportionally, and the triangle, formed by the joining lines, and the remaining side, shall be equal to a third of the original triangle. 67. Find the locus of a point, such that the distance between the feet of the perpendiculars from it upon two straight lines, given in position, may be constant. 68. ABCD is a parallelogram, AC, BD diagonals. If parallel lines be drawn through A, C, and also through B, D, the diagonals of all parallelograms so formed will pass through the same point. 69. OPQ is any triangle. OR bisects PQ in R; PST bisects OR in S, and cuts OQ in T. Shew that OQ=30T. 70. If the side BC, of a triangle ABC, be bisected by a line, which meets AB and AC, produced if necessary, in D and E respectively, shew that AE is to EC as AD is to DB. 71. Two circles are drawn in the same plane, having a common centre C. If the tangent, at any point P of the inner circle, meet the outer in Q, and be produced both ways to points A, B, such that QA, QB, are each of them equal to QC, the area of the triangle CAB will be constant. 72. From P, a point without a circle, whose centre is C, two tangents PA, PB, are drawn, and also a line, meeting the circle in D, and AB in E. If CF be perpendicular to PD, then FD is a mean proportional between FP and FE. 73. Three circles touch the sides of a triangle ABC in the points where the inscribed circle touches them, and touch each other, in the points G, H, K. Prove that AG, BH and CK meet in a point. 74. If ABC be a right-angled triangle, and EF, parallel to BC, the hypotenuse, meet AB, AC in E, F, then EH, FL, AK being drawn perpendicular to BC, shew that the difference of the rectangles CK, CH and BL, BK is equal to the difference of the squares on AB, AC. 75. From a point A in the circumference of a circle two chords AB, AC are drawn, cutting off arcs greater than a quadrant and less than a semicircle; and from the extremity B of the greater chord, a line BD is drawn in a direction perpendicular to that of the diameter through A, and meets AC produced in D. Shew that AD is to AB as AB is to AC. 76. Two circles intersect, and through a point of intersection two lines are drawn, terminated by the circumferences of both circles; one of these lines remains fixed, while the other may have any position. Shew that the locus of the intersection of the lines joining their extremities is a circle. 77. If the side BC of an equilateral triangle ABC be produced to any point D, and AD be joined, and if a straight line CE be drawn parallel to AB, cutting AD in E, prove that the square on AE is to the rect. DA, DE as the rect. CE, CB is to the square on DC. 78. In a triangle, right-angled at A, if the side AC be double of AB, the angle B is more than double the angle C. 79. From the obtuse angle of a triangle draw a line to the base, which shall be a mean proportional between the segments, into which it divides the base. 80. AB, AC are two straight lines, B and C given points in the same; BD is drawn perpendicular to AC, and DE perpendicular to AB; in like manner CF is drawn perpendicular to AB, and FG to AC. Shew that EG is parallel to BC. 81. AB is the diameter of a circle, and CD a chord at right angles to it, E any point in CD. If AE and BE be drawn and produced to cut the circle in F and G, the quadrilateral FCGD has any two of its adjacent sides in the same ratio as the remaining two. 82. ADEB is a semicircle; AB the diameter; DF, EG perpendiculars on the diameter; C the centre of a circle, which touches the semicircle and these perpendiculars; and CH is drawn perpendicular to the diameter, Shew that CH is a mean proportional to AF and BG, 83. Divide a straight line in a given ratio, and produce it so that the whole line thus produced shall be to the part produced in the same ratio; shew that the circle described on the line between the two points of section, as diameter, is such, that if any point of its circumference be joined with the extremities of the given line, the straight lines so drawn shall also be in the given ratio. 84. AB, CF, DE, are chords in a circle, intersecting in 0. CE, DF joined cut AB in G and H respectively. Shew that the rectangle AO, OB is to the rectangle GO, OH as the difference between 40 and OB is to the difference between GO and OH. 85. Triangles on the same base, and with equal vertical angles, are to one another as the products of their sides. 86. A line ACBD is divided, so that AC is to CB as AD is to DB. Shew that a semicircle, described on CD, is the locus of P, such that AP is to PB as AC is to CB. 87. If the two diagonals of a quadrilateral, inscribed in a circle, be given, shew that the quadrilateral is greatest, when they are at right angles. 88. ABC is a triangle, D, E, the middle points of AB, AC, and BE, CD, meet in F: a triangle is drawn, having its sides parallel to AF, BF, CF. Shew that the lines, joining its angular points to the middle points of its opposite sides, will be parallel to the sides of the triangle ABC. 89. A circle rolls within another, of twice its radius: if P be the point of contact, and ▲ a given point of the rolling circle, PA will be constant in direction. 90. Two circles intersect; the line AHKB joining their centres A, B, meets them in H, K. On AB is described an equilateral triangle ABC, whose sides BC, AC intersect the circles in F, E. FE produced meets BA produced in G. Shew that as GA is to GK, so is CF to CE, and so also is GH to GB. 91. If, in Euclid's construction for forming a triangle ABC, with each of the angles B and C double of the angle A, D is the point where AB is divided, and AE is taken in AB equal to BD, shew that the area AEC is equal to the area BDC. 92. An isosceles triangle has one of its equal sides a mean proportional between two sides of another triangle. If these two sides include the same angle as the vertical angle of the isosceles triangle, shew that the triangles are equal. 93. Two triangles ABC, BCD, have the side BC common, the angles at B equal, and the angles ACB, BDC right angles. Shew that the triangle ABC is to the triangle BCD as AB is to BD. 94. Given the straight line which is drawn from the vertex of an equilateral triangle to a point of trisection of the base, find the side of the triangle. 95. Straight lines being drawn from the angular points A, B, C, of a triangle through any the same point, so as to cut the opposite sides respectively in a, b, c, shew that the rectangle Ab, Bc is to the rectangle Ac, Ba as Cb is to Ca. 96. ABCD is a quadrilateral inscribed in a circle, and its diagonals intersect in F. Shew that the rectangle AF, FD is to the rectangle BF, FC as the square on HD is to the square on BC. 97. ABCD is a quadrilateral figure whose opposite angles are not supplemental; the circle described about ABD cuts DC in E, and the circle described about BCE cuts AE in F. Shew that the triangle ABF is equiangular to the triangle BCD, and the triangle BCF to the triangle ABD. 98. ACB is a triangle whereof the side AC is produced to D until CD is equal to AC; and BD is joined, shew that if any line drawn parallel to AB cuts the sides AC and CB, and from the points of section lines be drawn parallel to DB, these will meet AB in points equidistant from its extremities. 99. A and B are fixed points, and AC, BD are perpendiculars on CD, a given straight line: the straight lines AD, BC, intersect in E, and EF is drawn perpendicular to CD. Shew that EF bisects the angle AFB. 100. If O be the centre of a circle circumscribed about the triangle ABC, obtuse-angled at C, and if in OC a circle be described meeting AB in D and E, then either CD or CE shall be a mean proportional between the segments into which they respectively divide AB. 101. The exterior angle CBD of the triangle ABC is bisected by the line BE, which cuts the base produced in E. Shew that the square on BE, together with the rectangle AB, BC, is equal to the rectangle AE, EC. 102. ABCD is a quadrilateral figure inscribed in a circle; BA, CD, are produced to meet in P, and AD, BC, are produced to meet in Q. Prove that PC is to PB as QA is to QB. Also, shew that half the sum of the angles at P and Q is equal to the complement of the opposite angle ABC of the quadrilateral figure. 103. Having given the vertical angle, and the ratio of the sides containing it, and also the diameter of the circumscribing circle, construct the triangle. |