34. If the two angles, which one straight line makes with another on the same side of it, be bisected, shew that any line cutting four of the lines thus drawn, will be harmonically divided; and, conversely, if a line be divided harmonically, and from any point lines be drawn to the four points of section, of which any two alternate ones contain a right angle, then the angle between the other two will be bisected. 35. If from any point P without a given circle tangents PC, PD be drawn to it, and the line, CD, joining the points of contact, cut in Q the diameter AOB, which passes through P, shew that OP.OQ= (rad.)2; and that the line PB is harmonically divided, and the angle PCQ bisected.* 36. If in [35] any line be drawn through P cutting the circle in E, G, and CD in F, shew that PG is harmonically divided, and the angles PEQ, PGQ bisected by EA, GA. 37. If P be the pole of CQD, then the pole of any line through P will lie in CQD: and, conversely, the polar of any point in CQD will pass through P. 38. Apply [37] to shew that, if a circle inscribed in a quadrilateral, ABCD, touch its sides AB, &c. in E, F, G, H, and EH, FG, be produced to meet in K, then KO drawn to the centre is perpendicular to AC. 39. In the triangle ABC, AC=2BC: if CD, CE, bisect the angle C and the exterior angle formed by producing AC, then the triangles CBD, ACD, ABC, CDE, have their areas as 1, 2, 3, 4. 40. The locus of the vertices of all triangles, upon the same base and having their sides in a given ratio, is a circle. 41. Find the locus of points at which two given circles will be seen under the same angle. The point P is called the pole of CQD, and CQD the polar of P. 42. In any triangle, ABC, right-angled at A, bisect the angle C by CD, and shew that 2AC2: AC2-AD2 :: AB: AD. 43. Construct a triangle, having given one side, the angle opposite to it, and the ratio of the other two sides. 44. If two triangles have one angle of the one equal to one angle of the other and another angle of the one supplementary to another angle of the other, the sides opposite to these four angles will be proportionals. 45. Through any point in the line bisecting an angle, draw a line cutting the sides at equal angles: this is the shortest line that can be drawn through the given point to cut them, and the triangle so cut off is the least possible. 46. The line, which cuts at equal angles the lines containing a given angle, is the least that can be drawn to cut off a triangle of given area. 47. If an isosceles triangle be inscribed in a circle, and from the vertical angle a line be drawn to meet the circumference and base, the rectangle of the segments of this line is equal to the square of either of the sides of the triangle. 48. Within a given circle place six equal circles touching one another and the given circle; and shew that the interior circle, which touches them all, is equal to each of them. 49. The perpendiculars from the angles of a triangle upon the opposite sides will meet in a point. 50. AB is the diameter of a circle, AC, BD, are any two chords intersecting in E: draw EF perpendicular to AB, and shew that, if produced, it passes through the intersection of AD, BC. 51. Find a point without a given circle, such that the sum of the two lines, drawn from it touching the circle, shall be equal to the line drawn from it through the centre to meet the circle. 52. ABC is an isosceles triangle; draw AD perpendicular to the base, and DEF, cutting AB, AC, in E, F: then AD: DE :: AB+AF: AB-AF. 53. From any point P, tangents PA, PB, are drawn to a circle, and AC is drawn perpendicular to the diameter BD: shew that AC is bisected by PD in E. 54. Divide a given line into any number of equal parts; and thence shew how to divide a triangle into the same number of equal parts, by lines drawn from a point in one of its sides. 55. AD is drawn bisecting the vertical angle of a triangle, and cutting the base BC in D; in BC produced take a point E, equally distant from A and D, and shew that BE: DE::DE: CE. 56. If through the bisection of the base of a triangle any line be drawn, cutting one side of the triangle, the other produced, and a line drawn parallel to the base from the vertex, this line shall be cut harmonically. 57. If four diverging lines cut a straight line harmonically, they will cut any other intercepted line harmonically. 58. Let the two circles, radii R and r, which touch (i) the three sides of a triangle ABC, and (ii) one side BC and the other sides produced, touch AB in D1, D2, AC in E1, E2: then shew that BD1. BD2=CE1. CE2= Rr. 59. AD is drawn perpendicular on the hypothenuse BC of a right-angled triangle: if R be the radius of the circle inscribed in ABC, and r, r', of those in ABD, ACD, shew that R2=r2+r'2. 60. If the exterior angle of a triangle be bisected, by a line which cuts the base produced, the square of this line will be equal to the difference of the rectangles of the segments of the base and of the sides of the triangle. 61. If from the angle A of any parallelogram any line be drawn cutting the diagonal in E, and the sides BC, CD, in F, G, shew that AE is a mean proportional between EF, EG. 62. Given the nth part of a given line, find the (n+1)th part. 63. CAB, CEB, are two triangles which have a common angle B, and the sides CA, CE, equal: if, in BE produced, there be taken ED, a third proportional to BA, AC, then will the triangles BDC, BAC, be similar. 64. APB is the quadrant of a circle, SPT a tangent at P, cutting the radii OA, OB, in S, T: draw PM perpendicular to OA, and shew that the triangle SOT: AOB:: ACB: OMP. 65. Through a given point draw a line, which, if produced, would pass through the point of intersection of two given lines, without producing them to meet. 66. If two triangles are equal, and have the sides about one angle in each reciprocally proportional, these angles are either equal or supplementary to each other. 67. Construct an isosceles triangle equal to a given scalene triangle, and with the same vertical angle. 68. ABCD is a rectangle; draw any line AE to CD and BF perpendicular to AE, and shew that the rectangle AĒ, BF, is equal to the given rectangle, ABCD. 69. ABC is an inscribed triangle, AD, AE, lines drawn to the base, parallel to the tangents at B, C: shew that AD=AE, and BD : CE :: AB2: AC2. 70. In any triangle, if a perpendicular be dropped from the vertex on the base, the base: sum of sides :: diff. of sides: diff. or sum of segments of the base, according as the perpendicular falls within or without the triangle. 71. If any line ABCDE be drawn cutting two intersecting circles, C being the point in which it meets their line of section, then AB: BC:: ED: DC, and AE2: BD2 :: AC.CE: BC.CD. 72. If a rectangle be inscribed in a right-angled triangle, having the right-angle common, the rectangle of the segments of the hypothenuse will be equal to the sum of the rectangles of the segments of the sides. 73. If an isosceles triangle be inscribed in a circle, having each of the sides double of the base, shew that the square upon the radius: that upon one of the sides :: 4:15. 74. ABC is a triangle, right-angled at A, and having the angle B double of the angle C; draw BD bisecting the angle B, and AE, DF, perpendiculars on BC, and shew that 75. If a line touch a circle, and a perpendicular be drawn from the point of contact on any diameter, and if from the extremities of this diameter and from the centre perpendiculars to the diameter be drawn to meet the tangent, the four perpendiculars will be proportionals. 76. Inscribe a square in a given regular pentagon. 77. Let the lines AB, AC, be cut proportionally in D, E, and let perpendiculars at D, E, intersect in F: then shew that, for all such positions of D and E, F will lie in a fixed line through A. 78. AB is any chord of a circle; AC, BC, are drawn to any point in the circumference, and cut the diameter perpendicular to AB in D, E; if O be the centre, shew that OD.OE=(rad)2. 79. If semicircles be described on the segments of the hypothenuse made by a perpendicular from the vertex of a right-angled triangle, the segments of the sides intercepted by them will be in the triplicate ratio of the sides. 80. The sides containing a given angle are in a given ratio, and the vertex is fixed: shew if the extremity of one of the sides moves in a given line, so also does the extremity of the other. 81. If from the extremities of the base of a triangle two lines be drawn, each parallel to one of the sides and equal to the other, the lines joining their other |