21. To draw straight lines the squares on which shall be equal to two, three, four, etc. times the square on a given straight line.

[Draw a line equal to the given line at right angles to it and complete the triangle; the square on the hypotenuse is double the given square. From the end of the last hypotenuse draw a line at right angles to it and equal to the given line, and complete the triangle; the square on this second hypotenuse is three times the given square. And so on.]

22. Two lines are given in position; find the locus of a point equidistant from them.

[(1) If the lines are parallel, the locus is the line midway between them.

(2) If the lines are not parallel, produce them to meet; bisect the angle so formed and this line is the locus required; prove by Prop. 26, drawing perpendiculars to the two lines from any point in the bisector.]

III. Parallelograms.

23. The diagonals of a parallelogram bisect each other. [Prove by Props. 29 and 26.]

24. A parallelogram is bisected by any straight line which passes through the middle point of one of its diagonals.

[Prove by Props. 29 and 26.]

25. If two equal triangles ABC, ABD are on the same base AB, but on opposite sides of it, the line joining their vertices C, D is bisected by AB.

[Complete the parallelogram ADBE; CE is parallel to AB and DE is bisected by AB; therefore DC is bisected by AB by No. 4, p. 101.]

26. To inscribe a square in a given triangle.

[ABC is the triangle; draw AD perpendicular to BC; produce BC to E, making CE equal to BD; bisect the angle ADE by DF, meeting AE in F; through F draw FKL parallel to BC meeting AB, AC, AD in L, K, and H respectively; draw KM, LN perpendicular to BC; KMNL is the square required. Prove FH equal to HD, and FL equal to HK. This construction assumes that the angles at the base are acute.]

MISCELLANEOUS RIDERS ON BOOK I.

I.-Depending on the Equality of Triangles.

1. If the straight line joining the vertex of a triangle to the middle point of the base is perpendicular to the base, the triangle is isosceles.

2. If two straight lines bisect each other at right angles, any point in either is equidistant from the extremities of the other line.

3. Find a point which is equidistant from three given points. When is this impossible?

4. In a given straight line find a point equidistant from two given intersecting straight lines. Is this always possible?

5. Find a point equidistant from three given straight lines. When is this impossible?

6. Through a given point P draw a straight line making equal angles with two given intersecting straight lines. Can more than one such line be drawn?

7. AB and CD are two lines which meet in an inaccessible point, and AX is drawn at right angles to AB; find a point in AX equidistant from AB and CD.

8. AB and CD are two lines which meet in an inaccessible point, and P is any point between them; show how to draw a line through P, which, when produced, would pass through the intersection of AB and CD.

9. If it be possible within a quadrilateral ABCD, whose opposite sides are equal, to find a point 0, such that OA, OB, OC, OD are all equal, then AOC, BOD are straight lines and ABCD is equiangular.

10. If the straight line joining the middle points of two opposite sides of a quadrilateral be at right angles to each of these sides, the other two sides of the quadrilateral are equal.

11. If two of the medians of a triangle are equal the triangle is isosceles. 12. Prove by the method of superposition that if two right-angled triangles have their hypotenuses equal and two angles equal, the triangles are equal in all respects.

13. If there be two isosceles triangles on the same base, the straight line which joins their vertices, produced if necessary, bisects the common base, and is perpendicular to it.

14. ABCD is a trapezium, of which the side AB is parallel to the side DC. Show that the area of ABCD is equal to the area of a parallelogram formed by drawing through M, the middle point of BC, a straight line parallel to AD. 15. If the straight line, joining two opposite angles of a parallelogram, bisects the angles, the parallelogram is a rhombus.

16. If two triangles have two sides of the one equal to two sides of the other, each to each, and the sum of the angles contained by these sides equal to two right angles, the triangles shall be equal in area,

II.—On Inequalities.

17. The perpendicular is the shortest line that can be drawn from a given point to a given straight line; and of any other straight lines drawn from the given point to the given line, that which is nearer to the perpendicular is always less than one more remote.

18. From the same point two and only two straight lines can generally be drawn to a given straight line, each equal to another given straight line.

When can only one, or no such straight line be drawn?

19. ABC is a triangle having the side AB greater than the side AC, and AD is drawn perpendicular to BC; prove that BD is greater than CD, and the angle BAD greater than the angle CAD.

20. In any triangle ABC the sum of the sides AB, AC is greater than twice the median AX.

21. In any triangle an angle is acute, right, or obtuse, according as the median drawn from that angle is greater than, equal to, or less than half the side opposite the angle.

22. ABC is a triangle of which AX is the median, AD is the bisector of the vertical angle, and AP is the perpendicular on BC; if AB is greater than AC, prove that BX is less than BD, and BD less than BP.

23. Of all equal triangles on the same base the isosceles has the least perimeter.

24. If a polygon be not regular, there may be found another polygon having the same number of sides and the same area, but having a less perimeter.

25. The perimeter of a variable triangle inscribed in a fixed triangle is a minimum when the sides of the former triangle make equal angles with the sides of the latter triangle.

26. The lines joining the feet of the perpendiculars drawn from the angles to the opposite sides of the triangle, form an inscribed triangle whose area is a minimum.

27. If the angle between two adjacent sides of a parallelogram be increased, while their lengths do not alter, the diagonal through the point of intersection will decrease.

III. On the Properties of Triangles.

28. In any triangle if a perpendicular be drawn from one extremity of the base to the bisector of the vertical angle, (i) it will make with either of the sides containing the vertical angle an angle equal to half the sum of the angles at the base; (ii) it will make with the base an angle equal to half the difference of the base angles; and (iii) the line joining the middle point of the base to the foot of the perpendicular will be equal to half the difference of the sides of the triangle.

29. If the base BC of a triangle ABC be produced to D, the angle between the bisectors of the angles ABC, ACD is equal to half the angle BAC.

30. If the sides AB, AC of a triangle ABC be produced to D and E, the bisectors of the angles BAC, CBD, BCE are concurrent.

31. If the sides AB, AC of a triangle ABC be produced to D and E, and the bisectors of the angles CBD, BCE meet in I, prove that the angle B/C is equal to half the sum of the angles ABC, ACB.

32. If the bisectors of the angles of a triangle ABC meet in / and IF is drawn perpendicular to AB, show that AF is equal to the difference between the semi-perimeter of the triangle ABC and the side BC.

33. If the bisectors of two of the exterior angles of the triangle ABC meet in / and ID, IE, IF are drawn perpendicular to BC, CA, AB respectively, produced when necessary, show that AE and AF are each equal to the semiperimeter of the triangle ABC.

34. In any triangle the difference of the base angles is double the angle between the perpendicular to the base and the bisector of the vertical angle.

35. In the triangle ABC, AX is the median, AP is the perpendicular to BC, and E and F are the middle points of AC and AB respectively; prove that the angles FPE and FXE are each of them equal to BAC.

36. No two straight lines drawn from two angles of a triangle and terminated by the opposite sides can bisect one another.

37. The triangle formed by the three bisectors of the exterior angles of a triangle is such that the lines joining its vertices to the angles of the original triangle will be its perpendiculars.

38. Any straight line which is drawn from the vertex of a triangle to the base is bisected by the straight line which joins the middle points of the other two sides of the triangle.

39. The three straight lines which join the middle points of the sides of a triangle, divide it into four triangles which are identically equal.

40. A is the vertex of an isosceles triangle, and BA, one of the equal sides, is produced to D, so that AD is equal to AB or AC; join DC and show that DC is at right angles to BC.

41. If the straight line which bisects an exterior angle of a triangle is parallel to the base of the triangle, prove that the triangle is isosceles.

42. ABC is an isosceles triangle; DEF is a straight line perpendicular to the base BC, meeting AB in E and CA produced in F; show that the triangle AEF is isosceles.

43. In a right-angled triangle if one acute angle be double the other acute angle, the hypotenuse will be double one of the sides.

44. In a right-angled triangle if a perpendicular be drawn from the right angle to the hypotenuse, it will divide the triangle into two triangles which are equiangular to one another and to the whole triangle,

45. Each of the exterior angles of a regular polygon is equal to the angle of an equilateral triangle ; find the number of sides to the polygon.

46. In an isosceles triangle the sum of the perpendiculars drawn from any point in the base to the equal sides is constant, and is equal to the perpendicular drawn from either of the equal angles to the opposite side.

47. If the base of an isosceles triangle be produced to any point, the difference of the perpendiculars drawn from this point to the equal sides is constant.

48. The sum of the perpendiculars from any point within an equilateral triangle to the three sides is constant.

49. In any triangle, if a perpendicular be drawn from the vertical angle to the base, the difference of the squares on the two sides of the triangle is equal to the difference of the squares on the segments of the base.

50. ABC is any triangle, and lines AD, BE, CF are drawn, making the angles BAD, CBE, ACF all equal to one another; prove that these lines either meet in a point, or else form a triangle equiangular to the triangle ABC.

IV. On the Construction of Triangles, etc.

51. How many triangles can be formed by taking any three lines from lines 2, 3, 4, 5, 6, 7 inches long? Which of the triangles so formed will be rightangled?

52. Construct an equilateral triangle having given its altitude.

53. Construct a triangle having given the middle points of its sides.

54. Construct a triangle having given the feet of the perpendiculars from the opposite angles.

55. Construct a triangle having given the base, the vertical angle, and the sum or the difference of the other two sides.

56. Construct a triangle having given the base, the sum of the sides and the difference of the base angles.

57. Construct a triangle equiangular to a given triangle and having its perimeter equal to a given straight line.

58. Construct an equilateral triangle one of whose angular points is given, and the other two lie one on each of two given straight lines.

59. Construct an isosceles triangle having given the base, and the sum of one of the equal sides and the altitude.

60. Construct a right-angled triangle having given the hypotenuse and the difference of the sides.

61. Construct a right-angled triangle having one of the acute angles double of the other.

62. Construct a right-angled triangle having given the hypotenuse and the perpendicular upon it from the right angle.