parallel; if the points of contact divide the other two sides proportionally, they are equally inclined to the first two. 18. If two triangles, on the same base, have their vertices joined by a straight line, which meets the base, or the base produced, shew that the parts of this line, between the vertices of the triangles and the base, are in the same ratio to each other as the areas of the triangles. 19. From any point P, in the circumference of a circle, whose centre is O, perpendiculars PM, PN, are let fall on two radii OA, OB, and are produced both ways to meet the circumference of the circle in C, D, and the straight lines OA, OB, in E, F respectively. Shew that the three straight lines CD, MN, EF, are parallel to one another. 20. If the angles B, C, of the triangle ABC, be respectively equal to the angles D, E, of the triangle ADE, and the angles B, E, of the triangle ABE, to the angles D, C, of the triangle ADC, then these pairs of triangles shall be respectively equal to each other; and if BE, CD, intersect in F, the triangles BFD, CFE, shall also be similar. 21. If, from the extremities of the diameter of a semicircle, perpendiculars be let fall on any line cutting the semicircle, the parts intercepted between those perpendiculars and the circumference are equal. 22. In a given circle place a chord, parallel to a given chord, and having a given ratio to it. 23. ABC is an equilateral triangle. Through C a line is drawn at right angles to AC, meeting AB produced in D, and a line through A parallel to BC in E. Through K, the middle point of AB, lines are drawn respectively parallel to AE, AC, and meeting DE in F and G. Prove that the sum of the squares on KG and FG is equal to three times the square on FE. 24. Find a point in the base of a right-angled triangle produced such that the line drawn from it to the angular point opposite to the base, shall be to the base produced as the perpendicular to the base itself. 25. AB is a given straight line, and D a given point in it; it is required to find a point P, in AB produced, such that AP is to PB as AD is to DB. 26. If two circles touch each other externally, and parallel diameters be drawn, the straight line, joining the extremities of those diameters, will pass through the point of contact. 27. If two circles touch each other, and also touch a straight line; the part of the line, between the points of contact, is a mean proportional between the diameters of the circles. 28. Two circles touch each other internally, the radius of one being treble that of the other. Shew that a point of trisection of any chord of the larger circle, drawn from the point of contact, is its intersection with the circumference of the smaller circle. 29. If ABC be a right-angled triangle, and D any point in its hypotenuse AB, determine by a geometrical construction the point P, to which AB must be produced, so that PA is to PB as AD is to DB. 30. If a line touching two circles cut another line joining their centres, the segments of the latter will be to each other as the diameters of the circles. 31. If through the vertex of an equilateral triangle a perpendicular be drawn to the side, meeting a perpendicular to the base, drawn from its extremity, the line, intercepted between the vertex and the latter perpendicular, is equal to the radius of the circumscribing circle. 32. If on the diagonals of a quadrilateral as bases, parallelograms be described, equal to the quadrilateral, and each containing an angle equal to a given angle, find the ratio of their altitudes. 33. The opposite sides BA, CD of a quadrilateral ABCD, which can be inscribed in a circle, meet, when produced, at E; F is the point of intersection of the diagonals, and EF meets AD in G; prove that the rectangle EA, AB is to the rectangle ED, DC as AG is to GD. 34. If from the extremities of the diameter of a circle tangents be drawn, any other tangent of the circle, terminated by them, is so divided at its point of contact, that the radius of the circle is a mean proportional between the segments of the tangent. 35. If the sides of a triangle, inscribed in the segment of a circle, be produced to meet lines drawn from the extremities of the base, forming with it angles equal to the angle in the segment, the rectangle contained by these lines will be equal to the square on the base. 36. Describe a parallelogram, which shall be of a given altitude, and equal and equiangular to a given parallelogram. 37. Two circles touch each other internally at the point A, and from two points in the line joining their centres perpendiculars are drawn, intersecting the outer circle in the points B, C, and the inner circle in the points D, E. Shew that AB is to AC as AD is to AE. 38. Given of any triangle the base, and the point, where the line, bisecting the exterior vertical angle, cuts the base produced, find the locus of the vertex of the triangle. 39. Draw a line from one of the angles at the base of a triangle, so that the part of it cut off by a line drawn from the vertex parallel to the base, may have a given ratio to the part cut off by the opposite side. 40. Find the point in the base produced of a right-angled triangle, from which the line drawn to the angle opposite to the base shall have the same ratio to the base produced, which the perpendicular has to the base itself. 41. If the centres A, B, of two circles be joined, and P be the point in the line AB, from which equal tangents can be drawn to the circles; the tangent drawn from any point in a line, which passes through P at right angles to AB are all equal. 42. Construct a triangle, similar to a given triangle, and having its angular points upon three given straight lines, which meet in a point. 43. Let ABCD be any parallelogram, BD its diagonal. Then the perpendiculars, from A on BD, and from B and D upon AD and AB, shall all pass through a point. 44. If a quadrilateral be inscribed in a circle, its diagonals shall be to one another as the sums of the rectangles contained by the sides adjacent to their extremities. 45. A square is described on the base of an isosceles triangle, remote from the vertex. Prove that, if the vertex be joined to the corners of the square, the middle segment of the base will be to the outer one in double the ratio of the perpendicular on the base to the base. 46. The base AB of an isosceles triangle ABC is produced both ways to D and E, so that the rectangle AD, BE is equal to the square on AC. Shew that the triangles DAC, EBC, are similar. 47. If each of the angles at the base of an isosceles triangle be double of the angle at the vertex, shew that either side is a mean proportional between the perimeter of the triangle, and the distance of the centre of the inscribed circle from either end of the base. 48. Prove that, if the rectangle contained by the diagonals of a quadrilateral be equal to the sum of the rectangles contained by its opposite sides, the quadrilateral may be inscribed in a circle. 49. Draw a line parallel to one of the sides of a triangle, so that it may be a mean proportional between the segments into which it divides one of the other sides. 50. If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, shew that the line joining the points of bisection will be trisected by the sides. 51. ABC is an equilateral triangle, BC is produced to D, and CD is made equal to BC: CE is drawn at right angles to DCB, and at A the angle CAE is made equal to the angle DCA; DE, DA are drawn. Shew that the rectangle DA, CE is equal to the rectangle DE, AC together with the square on CB. 52. Two straight lines AB, CD, intersect in E. If when AC, BD are joined, the sides of the triangle ACE; taken in order, are proportional to those of the triangle DBE, taken in order, shew that A, C, B, D, lie on the circumference of the same circle. 53. If any triangle be inscribed in a circle, and from the vertex a line be drawn parallel to a tangent at either extremity of the base, this line will be a fourth proportional to the base and two sides. 54. If a triangle be inscribed in a semicircle, and a perpendicular be drawn from any point in the diameter, meeting one side, the circumference, and the other side produced; the segments cut off will be in continued proportion. 55. If ABCD be any quadrilateral figure inscribed in a circle, and BK, DL be perpendiculars on the diagonal AC, shew that BK is to DL as the rectangle AB, BC is to the rectangle AD, DC. 56. If a rectangular parallelogram be inscribed in a rightangled triangle, and they have the right-angle common, the rectangle, contained by the segments of the hypotenuse, is equal to the sum of the rectangles, contained by the segments of the sides about the right angle. 57. If from the vertex of an isosceles triangle a circle be described, with a radius less than one of the equal sides, but greater than the perpendicular from the vertex to the base, the parts of the base cut off by it will be equal. 58. Through a fixed point A on a circle, a chord AB is drawn, and produced to a point M, so that the rectangle contained by AB and AM is constant. Find the locus of M. 59. Having given a circle and a point, another point may be determined, such that the segments of any chord of the circle, drawn through either point, shall subtend, at the other point, angles which are either equal or supplementary. |