divided into the same number of triangles which are actually equal, each to each, so that the divided figures may be superposed.

19. If from the vertex of a triangle, two straight lines be drawn to the base, one bisecting the vertical angle, and the other bisecting the base, prove that the latter is the greater of the two straight lines.

20. If from the vertical angle of a triangle three straight lines be drawn, one bisecting the angle, another bisecting the base, and the third perpendicular to the base, the first is always intermediate in magnitude and position to the other two.

21. In a right-angled isosceles triangle, the lines drawn from any of the angles to the opposite angle of the square described upon the opposite side are all equal.

22. Shew that the perimeter of the triangle, formed by joining the feet of the perpendiculars dropped from the angles upon the opposite sides of a triangle, is less than the perimeter of any other triangle, whose angular points are on the sides of the first.

23. From a given point there can be drawn only two equal straight lines to a given line, one on each side of the shortest line; and the shortest line is the perpendicular.

24. A, B are two fixed points: if two straight lines AC, BC be drawn making a given angle C, prove that the straight line bisecting C passes through a fixed point, and determine the point geometrically.

25. From every point of a given straight line, the straight lines drawn to each of two given points on opposite sides of the line are equal: prove that the line joining the given points will cut the given line at right angles.

26. From a given point without the angle contained by two straight lines given in position, draw a straight line in such a direction that the part of it intercepted between the given point and the nearest straight line, shall be equal to the part intercepted between the two straight lines.

27. If a straight line be drawn from a given point, and making a given angle with a given straight line, its length and the points of its intersection with the given line are given.

28. Shew that if there be two rectilinear figures on the same base, one of which wholly envelopes the other, the perimeter of the enveloping figure is greater than the perimeter of the other.

29. Shew that we may draw to a point within a triangle two straight lines which shall be greater than the sides of the triangle, if one of these two straight lines be not terminated in the extremity of the base.

30. If parallel lines be defined to be "lines in the same plane which make the same angle with any straight line which meets them," prove the following propositions respecting them.

(a) The alternate angles are equal.

(b) The two interior angles on the same side of the cutting line are equal to two right angles.

(c) Parallel lines never meet however far they are produced.

31. Can it be properly predicated of any two straight lines that they never meet if indefinitely produced either way, antecedently to our knowledge of some other property of such lines which makes the porperty first predicated of them a necessary conclusion from it?

32. Lines which are perpendicular to parallel lines are also parallel. 33. If the line joining two parallel lines be bisected, all the lines

drawn through the point of bisection and terminated by the parallel lines are also bisected in that point.

34. A quadrilateral figure whose sides are equal will be a parallelogram.

35. If the opposite angles of a quadrilateral figure be equal the opposite sides will be equal and parallel.

36. The diagonals of a parallelogram bisect each other.

37. The perimeter of a square is less than that of any other parallelogram of equal area.

38. Any straight line which bisects the diagonal of a parallelogram will also bisect the parallelogram; and no straight line can bisect a parallelogram unless it cut or meet the opposite sides.

39. The diagonals of a square and of a rhombus bisect each other at right angles.

40. If from any point in the diagonal of a parallelogram straight lines be drawn to the angles, the parallelogram will be divided into two pairs of equal triangles.

41. ABCD is a parallelogram of which the angle C is opposite to the angle A. If through A any straight line be drawn, then the distance of C is equal to the sum or difference of the distances of B and of D from that straight line according as it lies without or within the parallelogram.

42. If in a parallelogram two lines be drawn parallel to adjacent sides, and meeting the other sides of the figure, the lines joining their extremities, if produced, will meet the diameter in the same point.

43. ABCD is a parallelogram; draw the diagonal BC, and from D draw DE at right angles to BC, then if perpendiculars be drawn from B and C, they shall intersect in the line DE, produced if necessary.

44. If ABCD be a parallelogram, and E any point in the diagonal AC, or AC produced; shew that the triangles EBC, EDC are equal.

45. If from a point without a parallelogram, lines be drawn to the extremities of two adjacent sides, and of the diagonal which they include. Of the triangles thus formed, that, whose base is the diagonal, is equal to the sum of the other two.

46. It is impossible to divide a quadrilateral figure (except it be a parallelogram) into equal triangles by lines drawn from a point within it to its four corners.

47. If of the four triangles into which the diagonals divide a trapezium, any two opposite ones are equal, the trapezium has two of its opposite sides parallel.

48. If two sides of a trapezium be parallel, the triangle contained by either of the other sides and the two straight lines drawn from its extremities to the bisection of the opposite sides is equal to half the trapezium.

49. The sum of the diagonals of a trapezium is less than the sum of any four lines which can be drawn to the four angles, from any point within the figure, except their intersection.

50. When the corner of a leaf of a book is turned down a second time, so that the lines of folding are parallel and equidistant, the space in the second fold is equal to three times that in the first.

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. Along the sides of a parallelogram ABCD taken in order, measure AA' = BB' = CC' = DD' ; the figure A'B'C'D' will be a parallelogram.

53. If the points of bisection of the sides of a triangle be joined, the triangle so formed shall be one-fourth of the given triangle.

54. Prove that two lines drawn to bisect the opposite sides of a trapezium will also bisect each other.

55. Upon stretching two chains AC, BD, across a field ABCD, I find that BD and AC make equal angles with DC, and that AC makes the same angle with AD, that BD does with BC; hence prove that AB is parallel to CD.

56. If a line intercepted between the extremity of the base of an isosceles triangle, and the opposite side (produced if necessary) be equal to a side of the triangle, the angle formed by this line and the base produced is equal to three times either of the equal angles of the triangle.

57. AD, BC are two parallel straight lines, cut obliquely by AB and perpendicularly by AC; BED is drawn cutting AC in E so that ED is equal to twice BA; prove that the angle DBC is equal to onethird of the angle ABC.

58. AB, BC, DE, EF are rods joined at B, F, E, and D, capable of angular motion in the same plane, and so placed that FBDE is a parallelogram. If, when the rods are in any given position, points A, G and C be taken in the same line, shew that these points will always be in the same line, whatever be the angle the rods make with each other.

59. If upon the sides of a triangle as diagonals, parallelograms be described, having their sides parallel to two given lines, the other diagonals of the parallelograms will intersect in a point.

60. Prove that the perimeter of an isosceles triangle is greater than that of an equal rectangle of the same altitude.

61. If the areas of any triangle and of a square be equal, the perimeter of the triangle will be the greater.

62. If from the extremities A and B of the base of any triangle ABC, and on the same side of it, two straight lines AD, BE be drawn perpendicular to the base, each being double the altitude of the triangle, and straight lines DF, EG be drawn from D and E to the middle points of AC, BC; the sum or difference of the triangles ADF, BEG will be equal to the triangle ABC according as the angles, at the base of the latter, are acute or one of them is obtuse.

63. The perimeter of an isosceles triangle is less than that of any other equal triangle upon the same base.

64.

Of all triangles having the same base and the same perimeter, that is the greatest which has the two undetermined sides equal.

bases

65. Of all triangles having the same vertical angle, and whose pass through a given point, the least is that whose base is bisected in the given point.

66. If from the base to the opposite sides of an isosceles triangle three straight lines be drawn, making equal angles with the base, viz. one from its extremity, the other two from any other point in it, these two shall be together equal to the first.

67. From the extremities of the base of an isosceles triangle straight lines are drawn perpendicular to the sides, the angles made by them with the base are each equal to half the vertical angle.

68. If each of the equal angles of an isosceles triangle be onefourth of the third angle, and from one of them a perpendicular be drawn to the base meeting the opposite side produced; then will the part produced, the perpendicular, and the remaining side, form an equilateral triangle.

69. If two sides of a triangle be given, the triangle will be greatest when they contain a right angle.

70. The area of any two parallelograms described on the two sides of a triangle is equal to that of a parallelogram on the base, whose side is equal and parallel to the line drawn from the vertex of the triangle to the intersection of the two sides of the former parallelograms produced to meet.

AE.

71. In the figure to Prop. 47, Book 1,

(a) If BG and CH be joined, those lines will be parallel.

(b) If FC and BK be joined, they will cut off equal portions AD,

(c) 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.

(d) Shew that AL, BK, CF, intersect each other in the same point. (e) Join GH, KE, FD, and prove that each of the triangles so formed equals the given triangle ABC.

(f) The sum of the squares of GH, KE, and FD will be equal to eight times the square of the hypothenuse.

(g) If the exterior angular points of the squares be joined, an irregular hexagon will be formed, whose area is equal to the area of the square described upon the hypothenuse of a right-angled triangle, one of whose sides is equal to the hypothenuse of the original triangle, and the other is equal to the sum of its sides..

72. A point is taken within a square, and straight lines drawn from it to the angular points of the square, and perpendicular to the sides; the squares on the first are double the sum of the squares on the last. Shew that these sums are least when the point is in the centre of

the

square.

73. If from the vertex of a plane triangle, a perpendicular fall upon the base or the base produced, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base.

74. If from the middle point of one of the sides of a right-angled triangle a perpendicular be drawn to the hypothenuse, the difference of the squares of the segments into which it is divided, is equal to the square of the other side.

75. If a straight line be drawn from one of the acute angles of a right-angled triangle, bisecting the opposite side, the square upon that line is less than the square upon the hypothenuse by three times the square upon half the side bisected.

76. If one angle of a triangle be equal to a right angle, and another equal to two-thirds of a right angle, prove from Euclid, Book 1, that the equilateral triangle described on the hypothenuse, is equal to the sum of the equilateral triangles described upon the sides which contain the right angle.

GEOMETRICAL EXERCISES ON BOOK II.

THEOREM I.

The square of the excess of one straight line above another is less than the squares of the two straight lines by twice their rectangle.

Let AB, BC be the two straight lines, whose difference is AC. Then the square of AC is less than the squares of AB and BC by twice the rectangle contained by AB and BC.

Constructing as in Prop. 4. Book 11, Because the complement AG is equal to GE, add to each CK,

therefore the whole AK is equal to the whole CE; and AK, CE together are double of AK; but AK, CE are the gnomon AKF and CK, and AK is the rectangle contained by AB, BC; therefore the gnomon AKF and CK are equal to twice the rectangle AB, BC.

But AE, CK are equal to the squares of AB, BC;

hence taking the former equals from these equals, therefore the difference of AE, and the gnomon AKF is equal to the difference between the squares AB, BC, and twice the rectangle AB, BC;

but the difference AE and the gnomon AKF is the figure HF which is equal to the square of AC.

Wherefore the square of AC is equal to the difference between the squares AB, BC, and twice the rectangle AB, BC.

THEOREM II.

If straight lines be drawn from each angle of a triangle bisecting the opposite side, four times the sum of the squares of these lines is equal to three times the sum of the squares of the sides of the triangle.

Let ABC be any triangle, and let AD, BE, CF be drawn from A, B, C, to D, E, F, the bisections of the opposite sides of the triangle: draw AE perpendicular to BC.