EXERCISES. 1. The least straight line that can be drawn to another straight line from a point without it, is the perpendicular to it: of others, that which is nearer to the perpendicular is less than one more remote ; and only two equal straight lines can be drawn, one on each side of the perpendicular. 2. Of the triangles formed by drawing straight lines from a point within a parallelogram to the several angles, each pair that have opposite sides of the parallelogram as bases, are half of it. 3. If, in proceeding round an equilateral triangle, a square, or any regular polygon, in the same direction, points be taken on the sides, or the sides produced, at equal distances from the several angles, a similar rectilineal figure will be formed by joining each point of section with those on each side of it. x 4. If the three sides of one triangle be perpendicular to the three sides of another, each to each, the triangles are equiangular. 5. A trapezoid, that is, a trapezium having two of its sides parallel, is equal to a triangle which has its base equal to the sum of the parallel sides, and its altitude equal to their perpendicular distance. 6. Given the segments into which the line bisecting the vertical angle, divides the base, and the difference of the angles at the base ; to construct the triangle, and compute the sides. 7. Within or without a triangle, to draw a straight line parallel to the base, such that it may be equal to the parts of the other sides, or of their continuations, between it and the base. 8. Given the perpendicular of a triangle, the difference of the segments into which it divides the base, and the difference of the angles at the base ; to construct the triangle. 9. In the figure for the 47th proposition of the first book of Euclid, prove that CF is perpendicular to AD. 10. The angle made by two chords of a circle, or by their continuations, is equal to an angle at the circumference standing on an arc equal to the sum of the arcs intercepted between the chords, if the point of intersection be within the circle, or to their difference, if it be without: (2.) the angle made by a tangent and a line cutting the circle, is equal to an angle at the circumference on an arc equal to the difference of the arcs intercepted between the point of contact and the other line: and (3.) the angle made by two tangents is equal to an angle at the circamference standing on arc equal to the difference of those into which the circumference is divided at the points of contact. Cor. If a tangent be parallel to a chord, the arcs between the point of contact, and the extremities of the chord are equal. 11. Given the sum of the perimeter and diagonal of a square; to construct it. 12. On a given straight line describe a square, and on the side opposite to the given line, describe equilateral triangles lying in opposite directions ; circles described through the extremities of the given line, and through the vertices of these triangles are equal. 13. To inscribe an equilateral triangle in a given square. 14. Given the angles and two opposite sides of a trapezium to construct it. 15. In a given circle to place two chords of given lengths, and inclined at a given angle. 16. In the figure in page 91, if AM, BM be joined, the angle AMB is half of ACB. 17. If, in proceeding in the same direction round any triangle, as in Exercise 3, points be taken at distances from the several angles, each equal to a third of the side, the triangle formed by joining the points of section is one-third of the entire triangle. 18. To describe a square having two of its angular points on the circumference of a given circle, and the other two on two given straight lines drawn through the centre. Show that there may be eight such squares. 19. Given the vertical angle of a triangle, and the segments into which the base is divided at the point of contact of the inscribed circle ; to describe the triangle. 20. If any three angles of an equilateral pentagon be equal, all its angles are equal. 21. Given two sides of a triangle, and the difference of the segments into which the third side is divided by the perpendicular from the opposite angle; to construct the triangle. 22. Given the vertical angle of a triangle, the line bisecting it, and the perpendicular; to construct the triangle. 23. From a given centre to describe a circle, from which a straight line, given in position, will cut off a segment containing an angle equal to a given angle. 24. In a given triangle to inscribe a semicircle having its centre in one of the sides. 25. Through three given points to draw three parallels, two of which will be equally distant from the one between them. 26. Given an angle of a triangle, and the radii of the circles touching the sides of the triangles into which the straight line bisecting the given angle divides the triangle ; to construct it. 27. In a rhombus to inscribe a square. 28. Given the lengths of the two parallel sides of a trapezoid, and the lengths of the other sides; to construct it. 29. Given one of the angles at the base, and the segments into which the base is divided at the point of contact of the inscribed circle; to describe the triangle. 30. To draw a tangent to a given circle, such that the part of it intercepted between the continuations of two given diameters may be equal to a given straight line. 31. To draw a tangent to a given circle, such that the part of it between two given tangents to the circle may be equal to a given straight line. 32. Given the vertical angle of a triangle, the difference of the sides, and the difference of the segments into which the line bisecting the vertical angle divides the base ; to construct the triangle. 33. Given any three of the circles mentioned in the scholium to the fourth proposition of the fourth book; to describe the triangle. 34. A straight line and a point being given in position, it is required to draw through the point two straight lines inclined at a given angle, and enclosing with the given line a space of given magnitude. 35. From two given straight lines to cut off equal parts, each of which will be a mean proportional between the remainders. 36. A square is to a regular octagon described on one of its sides, as 1 to 2(1 + v 2). 37. In a given triangle to inscribe a parallelogram of a given area. 38. In a given circle to inscribe a parallelogram of given area. 39. Through a given point between the lines forming a given angle, to draw a straight line cutting off the least possible triangle. 40. To divide a given straight line into two parts, such that the square of one of them may be double of the square of the other, or may be in any given ratio to it. 41. To produce a given straight line, so that the square of the whole line thus produced, may be double of the square of the part added, or in any given ratio to it. 42. Given the area of a right-angled triangle, and the sum of the legs; to construct it. 43. Given the area and the difference of the legs of a rightangled triangle ; to construct it. 44. Given one leg of a right-angled triangle, and the remote segment of the hypotenuse, made by a perpendicular from the right angle; to construct the triangle. 45. Given the base of a triangle, the vertical angle, and the side of the inscribed square standing on the base; to describe the triangle. 46. Given the base of a triangle, and the radii of the inscribed and circumscribed circles ; to construct the triangle. 47. To draw a chord in a circle, which will be equal to one of the segments of the diameter that bisects it. 48. In a given circle to draw a chord which will be equal to the difference of the parts into which it divides the diameter that bisects it. 49. If two sides of a regular octagon, between which two others lie, be produced to meet, each of the produced parts is equal to a side of the octagon, together with the diagonal of a square, described on the side. 50. The perimeter of a triangle is to the base, as the perpendicular to the radius of the inscribed circle. 51. From a given point without a given circle, to draw a straight line cutting the circle, so that the external and internal parts may be in a given ratio. 52. Each of the complements of the parallelograms, about the diagonal of a parallelogram, is a mean proportional between those parallelograms. 53. Given the ratio of two straight lines, and the difference of their squares ; to find them. 54. The square of the perimeter of a right-angled triangle is equal to twice the rectangle under the sum of the hypotenuse and one leg, and the sum of the hypotenuse and the other. 55. The quadrilateral formed by straight lines bisecting each pair of adjacent sides of a quadrilateral, is a parallelogram which is half of the quadrilateral; and straight lines, joining the points in which the sides of that parallelogram are cut by the diagonals of the primitive figure, form a quadrilateral similar to that figure and equal to a fourth of it. 56. From three given points not in the same straight line, to describe three circles each touching the other two. Show that this admits of four solutions. 57. To add a parallelogram to a rhombus, such that the whole figure may be a parallelogram similar to the one added. 58. A straight line being given in position, and a circle in magnitude and position, it is required to describe two equal circles touching one another, and each touching the straight line and the circle. 59. The squares of the diagonals of a quadrilateral are together double of the squares of the straight lines joining the points of bisection of the opposite sides. 60. In a given rhombus to inscribe a rectangle having its sides in a given ratio. 61. If, through the vertex and the extremities of the base of a triangle, two circles be described intersecting one another in the base or its continuation, their diameters are proportional to the sides of the triangle. 62. To draw a straight line cutting two given concentric circles, so that the parts of it within them may be in a given ratio. 63. From a given point, within a given circle, or without it, to draw two straight lines to the circumference, perpendicular to one another, and in a given ratio. 64. If a straight line be divided in extreme and mean ratio, the squares of the whole and the less part are together equal to three times the square of the greater. 65. If a straight line be cut in extreme and mean ratio, and be also bisected, the square of the intermediate part, and three times the square of half the line are equal to twice the square of the greater part. 66. If the hypotenuse of a right-angled triangle be given, the side of the greatest inscribed square, standing on the hypotenuse, is one-third of the hypotenuse. 67. To divide a given semicircle into two parts by a perpendicular to the diameter, so that the radii of the circles inscribed in them, may be in a given ratio. 68. To draw a straight line parallel to the base of a triangle, making a segment of one side equal to the remote segment of the other. 69. In the figure for the 11th proposition of the second book of Euclid, the square of the diameter of the circle, passing through the points F, H, D, is six times the square of the straight line joining FD. 70. Given the base, the area, and the sum of the squares of the sides of a triangle ; to construct it. 71. If, from the extremities of the hypotenuse of a right-angled triangle as centres, arcs be described passing through the right angle, the hypotenuse is divided into three segments, such that the square of the middle one is equal to twice the rectangle of the others. 72. On a given hypotenuse to describe a right-angled triangle, such that the difference between one leg, and the adjacent segment of the hypotenuse made by a perpendicular from the right angle, may be a maximum. 73. On a given hypotenuse to construct a right-angled triangle, Compare this with Appendix, Book I., Prop. XXVII., Schol. c2 |