Draw PGE parallel to AD or BC, and meeting AB in G, and DO in E; and join DG, GC.

Then the triangles CBP, CBG are equal: (1. 37.)
and taking the common part CBH from each,
the remainders PHB, CHG are equal.

Again, the triangles DAP, DAG are equal; (1. 37.) also the triangles DAG, AGC are equal, being on the same base AG, and between the same parallels AG, DC:

therefore the triangle DAP is equal to the triangle AGC: but the triangle PHB is equal to the triangle CHG, wherefore the triangles PHB, DAP are equal to AGC, CHG, or ACH, add to these equals the triangle APH,

therefore the triangles APH, PHB, DAP are equal to APH, ACH, that is, the triangles APB, DAP are together equal to the triangle PAC.

If the point P be within the parallelogram, then the difference of the triangles APB, DAP may be proved to be equal to the triangle PAC.

I.

8. Describe an isosceles triangle upon a given base and having each of the sides double of the base, without using any proposition of the Elements subsequent to the first three. If the base and sides be given, what condition must be fulfilled with regard to the magnitude of each of the equal sides in order that an isosceles triangle may be constructed?

9. In the fig. Euc. 1. 5. If FC and BG meet in H, then prove that AH bisects the angle BAC.

10. In the fig. Euc. 1. 5. If the angle FBG be equal to the angle ABC, and BG, CF, intersect in O; the angle BOF is equal to twice the angle BAC.

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

12. A line drawn bisecting the angle contained by the two equal sides of an isosceles triangle, bisects the third side at right angles.

13. If a straight line drawn bisecting the vertical angle of a triangle also bisect the base, the triangle is isosceles.

14. Given two points one on each side of a given straight line; find a point in the line such that the angle contained by two lines drawn to the given points may be bisected by the given line.

15. In the fig. Euc. 1. 5, let F and G be the points in the sides AB and AC produced, and let lines FH and GK be drawn perpendicular and equal to FC and GB respectively: also if BH, CK, or these lines produced meet in O; prove that BH is equal to CK, and BO to CO.

16. 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.

17. If A be the vertex of an isosceles triangle ABC, and BA be produced so that AD is equal to BA, and DC be drawn; shew that BCD is a right angle.

18. The straight line EDF, drawn at right angles to BC the base of an isosceles triangle ABC, cuts the side AB in D, and CA produced in E; shew that AED is an isosceles triangle.

19. In the fig. Euc. 1. 1. if AB be produced both ways to meet the circles in D and E, and from C, CD and CE be drawn; the figure CDE is an isosceles triangle having each of the angles at the base, equal to one fourth of the angle at the vertex of the triangle.

20. From a given point, draw two straight lines making equal angles with two given straight lines intersecting one another.

21. From a given point to draw a straight line to a given straight line, that shall be bisected by another given straight line.

22. Place a straight line of given length between two given straight lines which meet, so that it shall be equally inclined to each of them.

23. To determine that point in a straight line from which the straight lines drawn to two other given points shall be equal, provided the line joining the two given points is not perpendicular to the given line.

24. In a given straight line to find a point equally distant from two given straight lines. In what case is this impossible?

25. If a line intercepted between the extremity of the base of an isosceles triangle, and the opposite side (produced if necessary) he 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.

26. In the base BC of an isosceles triangle ABC, take a point D. and in CA take CE equal to CD, let ED produced meet AB produced in F, then 3.AEF-2 right angles + AFE, or = 4 right angles + AFE.

27. 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.

28. A straight line is drawn, terminated by one of the sides of an isosceles triangle, and by the other side produced, and bisected by the base; prove that the straight lines, thus intercepted between the

vertex of the isosceles triangle, and this straight line, are together equal to the two equal sides of the triangle.

29. In a triangle, if the lines bisecting the angles at the base be equal, the triangle is isosceles, and the angle contained by the bisecting lines is equal to an exterior angle at the base of the triangle.

30. In a triangle, if lines be equal when drawn from the extremities of the base, (1) perpendicular to the sides, (2) bisecting the sides, (3) making equal angles with the sides; the triangle is isosceles : and then these lines which respectively join the intersections of the sides, are parallel to the base.

II.

31. ABC is a triangle right-angled at B, and having the angle A double the angle C; shew that the side BC is less than double the side AB.

32. If one angle of a triangle be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.

33. If from the right angle of a right-angled triangle, two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle.

34. If the vertical angle CAB of a triangle ABC be bisected by AD, to which the perpendiculars CE, BF are drawn from the remaining angles: bisect the base BC in G, join GE, GF, and prove these lines equal to each other.

35. The difference of the angles at the base of any triangle, is double the angle contained by a line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex.

36. If one angle at the base of a triangle be double of the other, the less side is equal to the sum or difference of the segments of the base made by the perpendicular from the vertex, according as the angle is greater or less than a right angle.

37. 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.

38. From the vertex of a scalene triangle draw a right line to the base, which shall exceed the less side as much as it is exceeded by the greater.

39. Divide a right angle into three equal angles.

40. One of the acute angles of a right-angled triangle is three times as great as the other; trisect the smaller of these.

41. Prove that the sum of the distances of any point within a triangle from the three angles is greater than half the perimeter of the triangle.

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

43. If from the angles of a triangle ABC, straight lines ADE, BDF, CDG be drawn through a point D to the opposite sides, prove that the sides of the triangle are together greater than the three

lines drawn to the point D, and less than twice the same, but greater than two-thirds of the lines drawn through the point to the opposite sides.

44. In a plane triangle an angle is right, acute or obtuse, according as the line joining the vertex of the angle with the middle point of the opposite side is equal to, greater or less than half of that side.

45. If the straight line AD bisect the angle A of the triangle ABC, and BDE be drawn perpendicular to AĎ and meeting AC or AC produced in E, shew that BD = DE.

46. The side BC of a triangle ABC is produced to a point D. The angle ACB is bisected by a line CE which meets AB in E. A line is drawn through E parallel to BC and meeting AC in F, and the line bisecting the exterior angle ACD, in G. Shew that EF is equal to FG.

47. The sides AB, AC, of a triangle are bisected in D and E respectively, and BE, CD, are produced until EF= EB, and GD = DC; shew that the line GF passes through A.

48. In a triangle ABC, AD being drawn perpendicular to the straight line BD which bisects the angle B, shew that a line drawn from D parallel to BC will bisect AC.

49. If the sides of a triangle be trisected and lines be drawn through the points of section adjacent to each angle so as to form another triangle, this shall be in all respects equal to the first triangle.

50. Between two given straight lines it is required to draw a straight line which shall be equal to one given straight line, and parallel to another.

51. 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.

52. In the base of a triangle, find the point from which, lines drawn parallel to the sides of the triangle and limited by them, are equal.

53. In the base of a triangle, to find a point from which if two lines be drawn, (1) perpendicular, (2) parallel, to the two sides of the triangle, their sum shall be equal to a given line.

III.

54. In the figure of Euc. I. 1, the given line is produced to meet either of the circles in P; shew that P and the points of intersection of the circles, are the angular points of an equilateral triangle.

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

56. In the figure Fuc. 1. 1, if the sides CA, CB of the equilateral triangle ABC be produced to meet the circles in F, G, respectively, and if C' be the point in which the circles cut one another on the

other side of AB: prove the points F, C, G to be in the same straight line; and the figure CFG to be an equilateral triangle.

57. ABC is a triangle and the exterior angles at B and C are bisected by lines BD, CD respectively, meeting in D: shew that the angle BDC and half the angle BAC make up a right angle.

58. If the exterior angle of a triangle be bisected, and the angles of the triangle made by the bisectors be bisected, and so on, the triangles so formed will tend to become eventually equilateral.

59. If in the three sides AB, BC, CA of an equilateral triangle ABC, distances AE, BF, CG be taken, each equal to a third of one of the sides, and the points E, F, G be respectively joined (1) with each other, (2) with the opposite angles: shew that the two triangles so formed, are equilateral triangles.

IV.

60. Describe a right-angled triangle upon a given base, having given also the perpendicular from the right angle upon the hy potenuse.

61. Given one side of a right-angled triangle, and the difference between the hypotenuse and the sum of the other two sides, to construct the triangle.

62. Construct an isosceles right-angled triangle, having given (1) the sum of the hypotenuse and one side; (2) their difference.

63. Describe a right-angled triangle of which the hypotenuse and the difference between the other two sides are given.

64. Given the base of an isosceles triangle, and the sum or difference of a side and the perpendicular from the vertex on the base. Construct the triangle.

65. Make an isosceles triangle of given altitude whose sides shall pass through two given points and have its base on a given straight line.

66. Construct an equilateral triangle, having given the length of the perpendicular drawn from one of the angles on the opposite side. 67. Having given the straight lines which bisect the angles at the base of an equilateral triangle, determine a side of the triangle.

68. Having given two sides and an angle of a triangle, construct the triangle, distinguishing the different cases.

69. Having given the base of a triangle, the difference of the sides, and the difference of the angles at the base; to describe the triangle. 70. Given the perimeter and the angles of a triangle, to construct it.

71. Having given the base of a triangle, and half the sum and half the difference of the angles at the base; to construct the triangle.

72. Having given two lines, which are not parallel, and a point between them; describe a triangle having two of its angles in the respective lines, and the third at the given point; and such that the sides shall be equally inclined to the lines which they meet.

73. Construct a triangle, having given the three lines drawn from the angles to bisect the sides opposite.