12. Determine as a fraction of a right angle the interior angle of a regular octagon. 13. The difference between any two sides of a triangle is less than the third side. 14. Describe an equilateral triangle having its side equal to a given finite straight line, and having one of its angular points at a given point. 15. Describe the mode of proof called the Reductio ad absurdum and mention any instances of its use in the First Book of Euclid. 16. In a triangle ABC find a point P in the side AB such that if the parallelogram PBC be completed it shall be equal to the triangle ABC. 17. If isosceles right-angled triangles be described on the sides of a right-angled triangle as bases, prove that the triangle on the side subtending the right angle is equal to the triangles on the sides containing the right angle. 18. In which of Euclid's subsequent propositions is I. 6 first required? 19. Explain why Euclid's, or some equivalent postulate, is necessary, in addition to the definition of parallel straight lines, in order to conduct reasonings on the properties of parallel lines. 20. If any point P be joined to A, B, C, D, the angular points of a rectangle, the squares on PA and PC are together equal to the squares on PB and PD. 21. Show that the diameter of the circle employed in the fig. of II. 14 is half the perimeter of the rectangle employed. 22. AB is parallel to CD and unequal to it, and they are joined towards the same parts by the straight lines AC and BD. If AC is equal to BD, show that AD is equal to BC. 23. A straight line is divided into two parts. Show that the sum of the squares on the parts is least when the straight line is bisected. 24. ABCD is a quadrilateral, AC and BD being the diagonals. The triangle ADC is half the triangle ABC, and the triangle BCD is half the triangle BAD. Prove that two of the sides of the quadrilateral are parallel, and that one of these is double of the other. 25. Show how to divide a right angle into six equal angles. 26. ABC is a triangle. Find a point D on the side BC such that the perpendicular from D on AC may be equal to DB. 27. ABC is a triangle having a right angle at C. Find a point D in BC so that the difference of the squares on AB and AD may be equal to three times the difference of the squares on AD and AC. 28. AB, AC are two straight lines given in position. It is required to find in them two points, P and Q, such that the angle APQ may be equal to a given angle, and the difference of AP and PQ equal to a given length. 29. From a given point P without a given straight line AB draw a straight line making a given angle with AB. 30. OA, OB are two given straight lines meeting in O; and P is a given point in the angle AOB. Show how to draw a straight line through P so that the part intercepted between OA, OB may be bisected in P. 31. Divide a given straight line AB into two parts AC, BC, so that the rectangle contained by the whole AB and one of the parts BC may be equal to the square on AC. Show that the difference of the squares on AC, BC is equal to the rectangle contained by AC, BC. 32. From a point within a quadrilateral straight lines are drawn to the four angular points. Determine the position of the point so that the sum of the four straight lines may have its least value. 33. Divide a given quadrilateral into two parts of equal area by a straight line drawn through an angular point. 34. Draw a line DE parallel to the base BC of a triangle ABC, cutting AB in D and AC in E, so that DE shall be equal to the sum of BD and CE. 35. Show how to construct a rectangle which shall be equal to a given square (1) when the sum, (2) when the difference, of two adjacent sides is given. 36. Show that the perpendicular is the shortest straight line which can be drawn from a given point to a given line. 37. AB, CD are two parallel straight lines, and AD, BC intersect in the point O. Prove that if AO is equal to OD, BO is equal to OC. 38. The base BC of an equilateral triangle is produced to D, so that the produced part CD is equal to BC. Show that the square of AD is equal to three times the square of BC. 39. On the sides of an equilateral triangle ABC, equilateral triangles BCD, CAE, ABF are described, all external. Show that the figure DEF is another equilateral triangle. 40. If a polygon of 40 sides be constructed, all of whose angles are equal to each other, prove that each angle differs from two right angles by the tenth of a right angle. 41. ABC is a triangle, and CD is drawn from the angular point C perpendicular to the opposite side AB. If E be any point in CD, prove that the sum of the squares described on AC and EB is equal to the sum of the squares described on BC and AE. 42. If two straight lines AB and CD bisect one another at their point of intersection, prove that the figure ABCD is a parallelogram. 43. Of all rectangles of equal perimeter the square is the greatest. 44. Produce a given straight line so that the rectangle contained by the whole line thus produced and the part produced shall be equal to the square of the given line. 45. If AEB, CED are two straight lines cutting one another at E, and if EF bisect the angle AEC, and EG bisect the angle CEB, show that EF is at right angles to EG. 46. If the diameters of a parallelogram are at right angles to one another, the parallelogram is either a square or a rhombus. 47. Find a straight line of which the square shall be equal to half the square of a given line. 48. Show that upon any given straight line as base it is possible to construct an equilateral triangle. Can only one such triangle be constructed? and, if more than one, how does that appear on the face of your construction? 49. If each interior angle of a polygon contains 165 degrees, how many sides has the polygon? 50. Describe a square equal to a given parallelogram (1) if the parallelogram is rectangular, (2) if it is oblique. 51. What is meant by saying that a right angle is an angle of 90 degrees? 52. If the three sides of one triangle are respectively equal to the three sides of another, prove that the three angles of the first are respectively equal to the three angles of the second triangle. 53. Specify the figures whose properties are discussed in the 'First Book of Euclid.' What is the least number of triangles into which a rectilineal plane figure of any given number of sides (e.g. 5 or 7 sides) can be dissected? 54. Deduce from I. 32 that all the exterior angles of any polygon are together equal to four right angles. Understanding by a regular polygon one whose sides and angles are all equal, find the magnitude of each interior angle of a regular polygon of 72 sides. 55. AC is the longest side of the triangle ABC. Find in AC a point D such that the angle ADB shall be equal to twice the angle АСВ. 56. How would you show that two circles which have equal radii are equal? 57. Given two points A and B on opposite sides of a given unlimited straight line, but at unequal distances from it, find a point P in this straight line such that the difference between AP and BP may be the greatest possible. 58. Show that the rectangle contained by two lines is equal to the excess of the square of half their sum above the square of half their difference. 59. What different kinds of magnitude are considered in plane geometry? 60. If the two diameters of a parallelogram are equal to one another show that the parallelogram is right-angled. 61. If the straight line BD meet the straight line ABC in B, show that the lines bisecting the angles ABD, CBD are at right angles to one another. 62. Upon AB, BC, CA, sides of the triangle ABC, perpendiculars are drawn from a point D, meeting the side or the sides produced on E, F, G respectively. Prove that AE2+ BF2 + CG2 : BE2+ CF2+ AG2. = 63. ABC is a triangle of which the greatest angle is the angle at A. Prove that if a point D be taken in AB and a point E in AC, DE is less than BC. 64. If the sides BC, CA, AB of a right-angled triangle ABC are bisected in D, E, F, show that twice the squares of AD, BE and CF are together equal to three times the square on the hypotenuse. 65. If two given straight lines intersect, and a point be taken equally distant from each of them, prove that it lies on one or other of the two straight lines which bisect the angles made with one another by the given straight lines. 66. What fraction of a right angle is the exterior angle of a regular heptagon? 67. If a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines shall be parallel.' What supposition is tacitly assumed concerning Р the two straight lines' in this enunciation? Why is the proposition not necessarily true as it stands ? 68. Investigate the sum of the exterior angles of any plane convex polygon. 69. State the relation which connects the obtuse angle of an obtuse-angled triangle, the sides containing the obtuse angle, and the segment lying in either one of these sides produced, and intercepted between the perpendicular drawn from the opposite acute angle upon the side so produced, and the obtuse angle. 70. Let ABC be any obtuse-angled triangle, C the obtuse angle, AD, BE perpendiculars upon BC, AC produced and meeting these produced lines in D and E. Prove that the rectangle of BC, CD is equal to that of AC, CE. 71. Bisect the angle between two unlimited straight lines which intersect. Through a given point draw a straight line equally inclined to two given straight lines. How many such lines may be drawn? 72. Prove that the angle at the vertex of a triangle is less than, equal to, or greater than a right angle according as the line joining the vertex to the middle point of the base is greater than, equal to, or less than half the base. 73. The hypotenuse of three isosceles right-angled triangles form a right-angled triangle. Construct the figure, and prove that one of the given triangles is equal to the sum of the other two. 74. AOB, COD are two indefinite straight lines intersecting each other in the point O, and P is a given point in the plane of these lines. It is required to draw through the point P a straight line PXY, cutting AB in X and CD in Y, in such a manner that OX may be equal to OY. Can this problem be solved in more than one way ? 75. Construct an isosceles triangle whose base and angle at the vertex are given. 76. If through the middle point of a side of a triangle a straight line be drawn parallel to the base it will bisect the other side, and that part of it which is intercepted between the sides of the triangle will be equal to half the base. 77. If A be the point of intersection of the diagonals of a parallelogram, then every straight line drawn through A will meet opposite sides of the parallelogram in points equidistant from A, and will divide the gram into two equal parts. |