36. The area of any trapezium is half the rectangle contained by one of the diagonals of the trapezium, and the sum of the perpendiculars let fall upon it from the opposite angles.

37. If the middle points of the sides of a triangle be joined, the lines form a triangle whose area is one-fourth that of the given triangle.

38. If the sides of a triangle be such that they are respectively the sum of two given lines, the difference of the same two lines, and twice the side of a square equal to the rectangle contained by these lines, the triangle shall be right-angled, having the right angle opposite to the first-named side..

39. If a point be taken within a triangle such that the lengths of the perpendiculars upon the sides are equal, show that the area of the rectangle contained by one of the perpendiculars and the perimeter of the triangle is double the area of the triangle.

40. In the last problem, if O be the given point, and OD, OE, OF the respective perpendiculars upon the sides BC, AC, and AB, show that the sum of the squares upon AD, OB, and DC exceeds the sum of the squares upon AF, BD, and CD by three times the square upon either of the perpendiculars.

41. Having given the lengths of the segments AF, BD, CE, in Problem 40, construct the triangle.

42. Draw a line, the square upon which shall be seven times the square upon a given line.

43. Draw a line, the square upon which shall be equal to the sum or difference of two given squares.

44. Reduce a given polygon to an equivalent triangle.

45. Divide a triangle into equal areas by drawing a line from a given point in a side.

46. Do the same with a given parallelogram.

47. If in the fig., Euc. I. 47, the square on the hypothenuse be on the other side, show how the other two squares may be made to cover exactly the square on the hypothenuse.

48. The area of a quadrilateral whose diagonals are at right angles is half the rectangle contained by the diagonals.

49. Bisect a given triangle by a straight line drawn from one of its angles.

50. Do the same with a given rectilineal figure ABCDEF.

51. If from the angle A of a triangle ABC a perpendicular be drawn meeting the base or base produced in D, show that the difference of the squares of AB and AC is equal to the difference of the squares of BD and DC.

52. If a straight line join the points of bisection of two sides of a triangle, the base is double the length of this line.

53. ABCD is a parallelogram, and E a point within it, and lines are drawn through E parallel to the sides of the parallelogram, show that E must lie on the diagonal AC when the figures BC and DE are equal.

54. If AD, BE, CF, are the perpendiculars from the angular points

of the triangle ABC upon the opposite sides, show that the sum of the squares upon AE, CD, BF, is equal to the sum of the squares upon CE, BD, AF.

55. The diagonals of a parallelogram bisect one another.

56. Write out at full length a definition of parallelism, and then prove that the alternate angles are equal when a straight line meets two parallel straight lines.

57. ABCDE are the angular points of a regular pentagon, taken in order. Join AC and BD meeting in H, and show that AEDH is an equilateral parallelogram.

58. Having given the middle points of the sides of a triangle, show how to construct the triangle.

59. Show that the diagonal of a parallelogram diminishes while the angle from which it is drawn increases. What is the limit to

which the diagonal approaches as the angle approaches respectively zero and two right angles?

60. A, B, C, are three angles taken in order of a regular hexagon, show that the square on AC is three times the square upon a side of the hexagon.

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