21. If one side of a triangle be greater than another, the perpendicular on it from the opposite angle is less than the corresponding perpendicular on the other side.

22. If one side of a triangle be greater than another, the median drawn to it is less than the median drawn to the other.

23. If one side of a triangle be greater than another, the bisector of the angle opposite to it is less than the bisector of the angle opposite to the other.

24. The hypotenuse of a right-angled triangle, together with the perpendicular on it from the right angle, is greater than the sum of the other two sides.

25. The sum of the three medians is greater than three-fourths of the perimeter of the triangle.

26. Construct an equilateral triangle, having given the perpendicular from any vertex on the opposite side.

Construct an isosceles triangle, having given:

27. The vertical angle and the perpendicular from it to the base. 28. The perimeter and the perpendicular from the vertex to the base.

Construct a right-angled triangle, having given :

29. The hypotenuse and an acute angle.

30. The hypotenuse and a side.

31. The hypotenuse and the sum of the other sides.

32. The hypotenuse and the difference of the other sides.

33. The perpendicular from the right angle on the hypotenuse and a side.

34. The median, and the perpendicular from the right angle, to the hypotenuse.

35. An acute angle and the sum of the sides about the right angle. 36. An acute angle and the difference of the sides about the right angle.

Construct a triangle, having given :

37. Two sides and an angle opposite to one of them. Examine the cases when the angle is acute, right, and obtuse.

38. One side, an angle adjacent to it, and the sum of the other two sides.

39. One side, an angle adjacent to it, and the difference of the other

two sides.

40. One side, the angle opposite to it, and the sum of the other two

sides.

H

41. One side, the angle opposite to it, and the difference of the

other two sides.

42. An angle, its bisector, and the perpendicular from the angle on

the opposite side.

43. The angles and the sum of two sides.

44. The angles and the difference of two sides.

45. The perimeter and the angles at the base. 46. Two sides and one median.

47. One side and two medians. 48. The three medians.

Construct a square, having given: 49. The sum of a side and a diagonal. 50. The difference of a side and a diagonal.

Construct a rectangle, having given :

51. One side and the angle of intersection of the diagonals.

52. The perimeter and a diagonal.

53. The perimeter and the angle of intersection of the diagonals.

54. The difference of two sides and the angle of intersection of the diagonals.

Construct a m, having given:

55. The diagonals and a side.

56. The diagonals and their angle of intersection.

57. A side, an angle, and a diagonal.

58. Construct a ||m the area and perimeter of which shall and perimeter of a given triangle.

59. The diagonals of all the ||ms inscribed* in a given ||m intersect one another at the same point.

60. In a given rhombus inscribe a square.

61. In a given right-angled isosceles triangle inscribe a square.

62. In a given square inscribe an equilateral triangle having one of its vertices coinciding with a vertex of the square.

63. AA', BB', CC' are straight lines drawn from the angular points of a triangle through any point O within the triangle, and cutting the opposite sides at A', B',C'. AP, BQ, CR are cut off from AA', BB', CC', and = OA', OB', OC'. Prove ▲ A'B'C' =▲ PQR.

* One figure is inscribed in another when the vertices of the first figure are on the sides of the second.

64. On AB, AC, sides of ▲ ABC, the ||ms ABDE, ACFG are described; DE and FG are produced to meet at H, and AH is joined; through B and C, BL and CM are drawn || AH, and meeting DE and FG at L and M. If LM be joined, BCML is a m, and = || BE + || CG. (Pappus, IV. 1.) 65. Deduce I. 47 from the preceding deduction.

66. If three concurrent straight lines be respectively perpendicular to the three sides of a triangle, they divide the sides into segments such that the sums of the squares of the alternate segments taken cyclically (that is, going round the triangle) are equal; and conversely.

67. Prove App. I. 2, 3 by the preceding deduction.

68. If from the middle point of the base of a triangle, perpendiculars be drawn to the bisectors of the interior and exterior vertical angles, these perpendiculars will intercept on the sides segments equal to half the sum or half the difference of the sides.

69. In the figure to the preceding deduction, find all the angles which are equal to half the sum or half the difference of the base angles of the triangle.

70. If the straight lines bisecting the angles at the base of a triangle, and terminated by the opposite sides, be equal, the triangle is isosceles. Examine the case when the angles below the base are bisected. [See Nouvelles Annales de Mathématiques (1842), pp. 138 and 311; Lady's and Gentleman's Diary for 1857, p. 58; for 1859, p. 87; for 1860, p. 84; London, Edinburgh, and Dublin Philosophical Magazine, 1852, p. 366, and 1874, p. 354.]

Locr.

1. The locus of the points situated at a given distance from a given straight line, consists of two straight lines parallel to the given straight line, and on opposite sides of it.

2. The locus of the points situated at a given distance from the ○ce of a given circle consists of the ces of two circles concentric with the given circle. Examine whether the locus will always consist of two

ces

[The distance of a point from the circumference of a circle is measured on the straight line joining the point to the centre of the circle.]

3. The locus of the points equidistant from two given straight lines which intersect, consists of the two bisectors of the angles made by the given straight lines.

4. What is the locus when the two given straight lines are parallel?

5. The locus of the vertices of all the triangles which have the same base, and one of their sides equal to a given length, consists of the Oces of two circles. Determine their centres and the length of their radii.

6. The locus of the vertices of all the triangles which have the same base, and one of the angles at the base equal to a given angle, consists of the sides or the sides produced of a certain rhombus.

7. Find the locus of the centre of a circle which shall pass through a given point, and have its radius equal to a given straight line.

8. Find the locus of the centres of the circles which pass through two given points.

9. Find the locus of the vertices of all the isosceles triangles which stand on a given base.

10. Find the locus of the vertices of all the triangles which have the same base, and the median to that base equal to a given length.

11. Find the locus of the vertices of all the triangles which have the same base and equal altitudes.

12. Find the locus of the vertices of all the triangles which have the same base, and their areas equal.

13. Find the locus of the middle points of all the straight lines drawn from a given point to meet a given straight line.

14. A series of triangles stand on the same base and between the same parallels. Find the locus of the middle points of their sides.

15. A series of ms stand on the same base and between the same parallels. Find the locus of the intersection of their diagonals.

16. From any point in the base of a triangle straight lines are drawn parallel to the sides. Find the locus of the intersection of the diagonals of every ||m thus formed.

17. Straight lines are drawn parallel to the base of a triangle, to meet the sides or the sides produced. Find the locus of their middle points.

18. Find the locus of the angular point opposite to the hypotenuse of all the right-angled triangles that have the same hypoten

use.

19. A ladder stands upright against a perpendicular wall. The foot of it is gradually drawn outwards till the ladder lies on the ground. Prove that the middle point of the ladder has described part of the Oce of a circle.

20. Find the locus of the points at which two equal segments of a straight line subtend equal angles.

21. A straight line of constant length remains always parallel to itself, while one of its extremities describes the Oce of a circle. Find the locus of the other extremity.

22. Find the locus of the vertices of all the triangles which have the same base BC, and the median from B equal to a given length.

23. The base and the difference of the two sides of a triangle are given; find the locus of the feet of the perpendiculars drawn from the ends of the base to the bisector of the interior vertical angle.

24. The base and the sum of the two sides of a triangle are given; find the locus of the feet of the perpendiculars drawn from the ends of the base to the bisector of the exterior vertical angle.

25. Three sides and a diagonal of a quadrilateral are given: find the locus (1) of the undetermined vertex, (2) of the middle point of the second diagonal, (3) of the middle point of the straight line which joins the middle points of the two diagonals. (Solutions raisonnées des Problèmes énoncés dans les Éléments de Géométrie de M. A. Amiot, 7ème ed. p. 124.)