15. A given angle BAC is bisected; if CA be produced to G, and the angle BAG bisected, the two bisecting lines are at right angles. 16. Prove the second part of Prop. I. 16, viz., that the exterior angle ACD is greater than the interior opposite angle ABC. 17. In the figure of Prop. I. 17, show that ABC and ACB are together less than two right angles, by joining A to any point in BC. 18. The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line; and of others, that which is nearer to the perpendicular is less than the more remote ; and two, and only two, equal straight lines can be drawn from the given point to the given straight line, one on each side of the perpendicular. 19. The difference of any two sides of a triangle is less than the third. 20. The three sides of a triangle taken together are greater than the double of any one side, but less than the double of any two sides. 21. To make a triangle equal to a given triangle. 22. The perpendiculars let fall on two sides of a triangle from any point in the straight line bisecting the angle between them, are equal to each other. 23. If two lines bisect perpendicularly two sides of a triangle, the perpendicular from their point of section upon the base, will bisect it. 24. If a straight line falling upon two other straight lines, make the two exterior angles on the same side of it equal to two right angles, these two straight lines are parallel. 25. Any straight line parallel to the base of an isosceles triangle makes equal angles with the sides. 26. If a line be perpendicular to one of two parallel lines, it will be perpendicular also to the other. 27. Lines which are perpendicular to parallel lines are also parallel. 28. The straight line which bisects the sides of a triangle is parallel to its base. 29. If a line be drawn through the middle point of one side of a triangle, parallel to the base, prove that it will bisect the other side. 30. Prove that the triangle cut off by this parallel is one-fourth part of the whole triangle. 31. Prove that the line drawn from the vertex of a triangle, bisecting the base, bisects every parallel to the base, whether above or below the vertex. 32. Each angle of an equilateral triangle is a third of two right angles, or two-thirds of one. 33. Trisect a right angle. 34. If the straight line bisecting the exterior angle of a triangle be parallel to the base, show that the triangle is isosceles. 35. If the sides of an equilateral and equiangular hexagon, or six-sided figure, be produced till they meet, the angles formed at the points of meeting are together equal to four right angles. 36. The parts of all perpendiculars to two parallel lines, intercepted between them, are equal. 37. If the opposite sides of a quadrilateral be equal, it is a parallelogram. 38. The diagonals of a parallelogram bisect each other. 39. The parallelogram, whose diameters are equal, is rectangular. 40. ABCD is a parallelogram, E is the middle point of DC; show that the triangle AEB is half the parallelogram. 41. Prove that the four triangles into which a parallelogram is divided by its diagonals are equal in area. 42. If the opposite sides of a trapezium be parallel to one another, the straight line joining their bisections, bisects the trapezium. 43. If, from any point within a parallelogram, straight lines be drawn to the extremities of two opposite sides, the two triangles upon these sides are together equal to half of the parallelogram. 44. The parallelograms about the diagonal of a square are squares. 45. In the figure, 1. 43, join EH, BD, and GF; prove that the three diagonals thus drawn, are parallel to each other. 46. In the figure, 1. 47, if BG and CH be joined, these lines will be parallel. 47. In the figure, 1. 47, if G and H be joined, show that the triangle GAH will be equal to the given triangle ABC. 48. In a rhombus, the squares of all the sides are together equal to the squares of the diagonals. 49. A straight line is drawn bisecting a parallelogram ABCD, and meeting AD in E and BC at F; show that the triangles EBF and CED are equal. 50. Given the diagonal of a square to construct it. 51. If the sides of a quadrilateral figure be bisected, and the points of bisection joined, the included figure is a parallelogram, and equal in area to half the original figure. 52. The square on the diameter of any square is double of it. 53. The square on the base of an isosceles triangle, whose vertical angle is a right angle, is equal to four times the area of the triangle. 54. If the diagonals of a parallelogram are equal, all its angles are equal. 55. Through a given point P, to draw a straight line, which shall cut off equal parts from two straight lines AB and AC, cutting one another in A. 56. To draw a straight line from a given point to meet another straight line, which shall make with it an angle equal to a given rectilineal angle. 57. To draw a straight line through a given point such that the part of it intercepted between two given parallel straight lines may be of given length. 58. Draw a line EF parallel to the base BC of a triangle ABC, so that EF shall be equal to BE. 59. In a given square to inscribe an equilateral triangle, having one of its angular points upon one of the angular points of the square, and its two remaining angular points one in each of two adjacent sides of the square. 60. To inscribe a square in a given right-angled isosceles triangle. 61. Given the perpendicular and the side of an isosceles triangle, to construct it. 62. If the two exterior angles at the base of a triangle be bisected, and the bisecting lines produced until they intersect, the line drawn from the point to the vertical angle will bisect it. 63. ABC is a triangle, right-angled at A, and having the angle B double of the angle C; show that the side CB is double of the side AB. 64. The lines which bisect the angles of any parallelogram, form a rectangular parallelogram, whose diameters are parallel to the sides of the former. 65. In any triangle ABC, if BE, CF be perpendiculars on any line through A, and D be the bisection of BC, show that DE=DF. 66. In any right-angled triangle, the middle point of the hypotenuse (side opposite the right angle) is equally distant from the three angles. 67. On the sides AC, BC of a triangle ABC, squares ACDE, BCFH are described; show that the straight lines AF and BD are equal. 68. The square on the side subtending an acute angle of a triangle is less than the squares on the sides containing the acute angle. 69. The square on the side subtending an obtuse angle of a triangle is greater than the squares on the sides containing the obtuse angle. 70. In the figure of Prop. 1. 47 prove that if perpendiculars be let fall from F and K on BC produced, the parts produced will be equal; and the perpendiculars together will be equal to BC. 71. If two circles cut each other, the line joining their points of intersection is bisected at right angles by the line joining their centres. 72. Describe a circle which shall pass through two given points, and have its centre in a given line. 73. From two given points on the same side of a given line, draw two lines which shall meet in that line and make equal angles with it. 74. Through a given point draw a line, so that the perpendiculars upon it from two other given points may be equal to each other. 75. In a given straight line, find a point equally distant from two given points; one in, and the other without, the given straight line. 76. If the straight lines bisecting the angles at the base of an isosceles triangle be produced to meet, they will contain an angle equal to an exterior angle of the triangle. 77. If in a right-angled triangle a line be drawn dividing the right angle into two parts, which shall be respectively equal to the adjacent base angles, prove that it will bisect the hypotenuse. 78. If a line be drawn from the middle of the hypotenuse of a right-angled triangle to the right angle, prove that it will be equal to half the hypotenuse. 79. Draw a line DE, parallel to the base BC of a triangle ABC, so that DE is equal to the sum of BD and CE. 80. Draw a line DE, parallel to the base BC of a triangle ABC. so that DE is equal to the difference of BD and CE. QUESTIONS AND EXERCISES ON BOOK II. DEFINITIONS. 1. Define a rectangle. By what is it contained? 2. Define a gnomon. PROPOSITIONS AND COROLLARIES. Prop. 1. If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. Prop. 2. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. Prop. 3. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part. Prop. 4. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. Cor. The parallelograms about the diameter of a square are likewise squares. Prop. 5. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. Cor. The difference of the squares of two unequal lines is equal to the rectangle contained by their sum, and their difference. Prop. 6. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced. Prop. 7. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Prop. 8. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, equal to the square of the straight line, which is made up of the whole and that part. Prop. 9. If a straight line be divided into two equal, and also into two unequal parts, the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section. Prop. 10. If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced. . Prop. 11. To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. Prop. 12. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. Prop. 13. In every triangle, the square of the side subtending either of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the acute angle and the perpendicular let fall upon it from the opposite angle. Prop. 14. To describe a square that shall be equal to a given rectilineal figure. ALGEBRAICAL AND ARITHMETICAL PROOFS OF THE The propositions of this book can be demonstrated both algebraically and arithmetically. A line may be represented by the number of linear units it contains; thus, the line AB, which contains a or 6 units, may be designated as a or 6. A rectangle, which in geometry is said to be contained by any two of the straight lines containing one of the right angles, may be expressed by the product of its length and breadth; thus, the rectangle contained by AB, AD, if AB be equal to a or 4, and AD to b or 3, the area of the rectangle may be represented by ab or 4×3. And therefore if the two sides of the rectangle be equal, or if b be equal to a, the figure is a square, which may be represented by multiplying the length of the side into itself, as axa-a2, or 4x4=42 or 16. An algebraical and arithmetical proof of each proposition will now be given, and the learner may be exercised in solving the problems independently by means of other letters and figures. PROP. I. THEOREM. Algeb.-Referring to the figure in Book II., if A=x, BC=a, BD=m, DE=n, and EC=p; then a=m+n+p, multiply both sides by x, ..ax=mx+nx+px. i.e., A.BC=A.BD+A. DE+A. EC. |