15. If an angle of a triangle and its opposite side be bisected by a straight line; the line is perpendicular to the side, and the triangle is isosceles.

16. Trisect a given straight line.

17. Trisect a right angle.

18. The greatest diameter of a parallelogram is that which is opposite to the greatest angle.

19. If the opposite sides of a quadrilateral figure are equal, it is a parallelogram.

20. If the opposite angles of a quadrilateral figure are equal, it is a parallelogram.

21. The difference between the angles at the base of a triangle is equal to twice the angle made by the straight line which bisects the remaining angle and the perpendicular from its vertex to the base.

22. From a given point in a side of a parallelogram, to draw a straight line that will bisect the parallelogram.

23. From a given point in the side of a triangle, to draw a straight line that will bisect the triangle.

24. Equal triangles between the same parallels, are upon equal bases.

25. If a straight line which is parallel to either side of a triangle bisect one of the other sides, it will also bisect the remaining side.

26. In figure to Prop. 39, if FD, GH, KE, be joined, the triangles FBD, GAH, KCE, are each equal to the triangle ABC.

27. If the square described upon one of the sides of a triangle be equal to the sum of the

squares described upon the other two sides of it; the angle contained by these two sides is a right angle.

28. One side of a right-angled triangle is 543, and the other side 367; find the hypothenuse by means of Prop. 39.

29. One side of a right-angled triangle is 645, and the hypothenuse 824; find the remaining side by the same Prop.

30. If the sides of a right-angled triangle be a and b, and the hypothenuse h, show alge

braically that a=(h+b) (h−b)3.

31. The sum of the 4 lines drawn from a point within a trapezium to the 4 angles, is the least, when that point is the intersection of the diameters.

32. If one angle of a triangle be equal to the other two together, it is a right angle.

33. The diameters of a rhombus bisect each other at right angles.

34. The diameters of a parallelogram bisect each other.

35. If two exterior angles of a triangle be bisected, and from the point of intersection of the bisecting lines, a line be drawn to the opposite angle of the triangle, it will bisect that angle. 36. Inscribe a square in a given rightangled isosceles triangle.

37. Inscribe a square in a given quadrant of a circle.

38. Inscribe a circle in a given quadrant of a circle.

39. If two straight lines cut each other at

right angles, and their extremities be joined, the sums of the squares of the opposite sides of the trapezium thus formed, are equal to each other.

40. The diameter is the greatest straight line that can be drawn in a circle.

41. If a perpendicular be drawn to a side of a triangle from the opposite angle; the rectangle of the sum and difference of the segments of that side, is equal to the rectangle of the sum and difference of the other sides.

42. The sides of a triangle are 3, 5, and 7; determine the numerical values of the perpendicular upon the longest side from the opposite angle, and also the segments of that side.

43. In any triangle, if a line from the middle of one side to the opposite angle be drawn; double the sum of the squares of the line, and of half that side is equal to the sum of the squares of the other two sides.

44. If two points be taken in the diameter of a circle equally distant from the centre; the sum of the squares of two lines drawn from these points to any point in the circumference will always be the same.

45. The sum of the squares of the two diameters of a parallelogram, is equal to the sum of the squares of the four sides.

46. The square of either side of an equilateral triangle is equal to three times the square of the radius of the circumscribing circle.

47. If two circles cut each other, they cannot have the same centre.

48. Perpendiculars from the centre of a circle to equal straight lines within the circle, are equal; and if the perpendiculars are equal, the lines are equal.

49. Draw a tangent to two given unequal circles.

50. Tangents to a circle from the same point, are equal to each other.

51. In equal circles, the angles which stand upon equal arcs are equal.

52. The difference between the hypothenuse and the sum of the other two sides of a right-angled triangle, is equal to the diameter of the circle inscribed in the triangle.

53. Inscribe a circle in a given triangle.

54. If an angle of a triangle be divided into two equal angles, by a straight line which cuts a side; the segments of that side will have the same ratio which the other sides of the triangle have to each other.

55. The four triangles into which a trapezium is divided by its diameters, are proportionals.

56. The triangle cut off by joining the middle points of the sides of a triangle, is onefourth of the whole.

57. The line which joins the middle point of the hypothenuse of a right-angled triangle and the right angle, is equal to half the hypothenuse.

58. Divide a given straight line in the same proportion as another given divided line.

59. Find a fourth proportional to three given straight lines.

60. Find a mean proportional between two given straight lines.

61. Bisect a triangle by a line drawn parallel to one of its sides.

62. The sum of the perpendiculars drawn from any point within an equilateral triangle to the sides, is equal to the perpendicular from either angle to its opposite side.

63. If from the extremities of any chord in a circle, perpendiculars be drawn meeting a diameter; the points of intersection are equally distant from the centre.

64. The perpendiculars from the middle points of the sides of a triangle, will all meet in a point.

65. The perpendiculars from the angles of a triangle to the opposite sides, will all meet in a point.

66. If from the three angles of a triangle lines be drawn to the middle points of the opposite sides, they will all meet in one point.

67. The three straight lines which bisect the angles of a triangle, meet in a point.

68. If from the three angles of a triangle lines be drawn to the points of bisection of the opposite sides; the squares of the distances between the angles and the common intersection, are together equal to one-third of the sum of the squares of the sides of the triangle.

69. If the diameter of a given circle be divided into any number of parts, on which circles are described, the circumferences of the latter will be together equal to the circumference of the given circle.