EXERCISES

ON THE

DIFFERENT PROPOSITIONS

OF THE

SIX BOOKS OF THE ELEMENTS.*

BOOK I.

1. At a given point in a straight line, there cannot be more than one perpendicular to the line, on the same side of it.

PROP. IV.

2. The straight line which bisects the vertical angle of an isosceles triangle bisects the base perpendicularly.

3. If two four-sided rectilineal figures have three consecutive sides of the one equal respectively to three consecutive sides of the other, and, likewise, the angles contained by the equal sides equal in each, the figures shall be equal in every respect. 4. If two equal triangles have one side, and an adjacent angle in the one, equal to one side and an adjacent angle in the other, the remaining sides and angles shall be equal, each to each.

ས.

5. If the base of an isosceles triangle be produced both ways, the exterior angles which it makes with the equal sides shall be equal.

6. The diameters of a rhombus bisect one another at right angles.

VII.

7. Prove that two circles, whose centres are given, do not cut each other in more than two points.

The Roman Numerals indicate the Proposition by means of which the Exercise is to be solved. It may be necessary to add, that although the exercise is generally appropriate to the proposition under which it is placed, there are some cases in which it is more appropriate to a preceding proposition, although requiring the aid of that under which it is placed; but in no case does the solution require the aid of any subsequent proposition

VIII.

8. If from the middle points of the three sides of a triangle, three straight lines be drawn at right angles to the sides, they shall all meet in the same point: and that point is equidistant from the three angular points of the given triangle.

XI.

9. In a straight line given in position, of indefinite length, to find a point which shall be equidistant from each of two given points. + Is this always possible?

10. To describe a circle which shall pass through three given points. Is this always possible?

XIV.

11. Only one straight line can be drawn at right angles to another from a given point without it.

XV.

12. In a given straight line to find a point such that the straight lines drawn from it to two given points shall make equal angles with the given straight line.

XIX.

13. Of all the straight lines which can be drawn to a given straight line from a given point without it, the perpendicular is the least; and of the rest, that which is nearer to the perpendicular is always less than one more remote: and there cannot be drawn more than two equal straight lines from the given point to the given line.

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

XX.

15. Any side of a triangle is greater than the difference between the other two sides.

16. The three sides of any triangle are together greater than the double of any one side; and less than the double of any two sides.

17. Any side of a polygon is less than the sum of the other sides. 18. The sum of two sides of a triangle is greater than double the straight line drawn from the vertex to the middle of the base. 19. The sum of two sides of a triangle is greater than double the straight line drawn from the vertex to the base, bisecting the vertical angle.

DEFINITION. The distance of a point from a straight line is the length of the perpendicular drawn from the point to the line. 20. If the point A be equidistant from the point B and the straight

line CD, any point in AB is nearer to B than to the straight line, but any point in BA produced from B to A is further from B than from the straight line.

21. In a given straight line to find a point such that the sum of two straight lines drawn to it from two points without the given line, shall be less than the sum of any two lines drawn from the same points, and terminated at any other point in the same line.

22. In two given straight lines to find two points such that the three straight lines which join them with two given points without the lines respectively, and with each other, shall be the least possible.

23. To determine a point in a line given in position to which lines drawn from two given points which are at unequal distances from the given line, may have the greatest difference possible.

XXI.

24. If a trapezium and a triangle stand upon the same base, and on the same side of it, and the one figure fall within the other, that which has the greater surface shall have the greater peri

meter.

XXVI.

25. Through a given point to draw a straight line which shall make equal angles with each of two given straight lines. 26. Through a given point to draw a straight line such that the segments, intercepted by perpendiculars let fall upon it from two given points, shall be equal.

XXVII.

27. A rhombus and a rhomboid are both parallelograms.

XXVIII.

28. Every rectangular four-sided figure is a parallelogram.

XXIX.

29. If two straight lines be respectively parallel to two others, the angle contained by the first two is equal to the angle contained by the other two.

30. To trisect a right angle.

XXXI.

31. Of all triangles which have the same vertical angle, and whose bases pass through the same point, the least is the one whose base is bisected in that point.

32. Through a given point between two given straight lines, to draw a straight line to meet them and be bisected at the given point.

XXXII.

33. A circle described from the point of bisection of the hypotenuse of a right-angled triangle as a centre, at the distance of half the hypotenuse, will pass through the summit of the right angle.

34. If the opposite angles of a quadrilateral figure be equal to one another, the figure is a parallelogram.

35. If two straight lines which cut one another be respectively perpendicular to two others which cut one another, the angles contained by the first two are respectively equal to the angles contained by the others.

36. If from the extremities of the base of a triangle, two segments be cut off, each equal to its adjacent side, and straight lines be drawn from the vertex to the points of section, these straight lines will contain an angle equal to half the sum of the angles at the base of the triangle.

37. If three straight lines be drawn, making equal angles with the three sides of a triangle, towards the same parts, they will form a triangle equiangular with the given triangle.

38. If four points be taken at equal distances from the angular points of a square, the figure which is formed by joining them is also a square.

39. If two straight lines which meet one another be cut by a third, and from the points of section two other straight lines be drawn, making with the first two the same angles which the cutting line makes with them respectively, the angle contained by the last two lines shall be double of the angle contained by the first two. 40. The hypotenuse of a right-angled triangle, together with the perpendicular on it from the right angle, are greater than the other two sides of the triangle.

XXXIV.

+41. The diagonals of a parallelogram bisect each other, and conversely.

42. If one side of a triangle be bisected, and through the point of bisection a straight line be drawn parallel to another side, this straight line bisects the third side; and conversely.

43. COR. The quadrilateral formed by joining the points of bisection of the sides of a quadrilateral is a parallelogram.

44. To draw a straight line terminated by the sides of a given tri+ angle, which shall be equal to one given straight line, and parallel to another.

45. In the base of a triangle to find a point from which lines drawn to each side of the triangle, and parallel to the other, shall be equal. ~46. If from any point in the base of an isosceles triangle, two straight lines be drawn, making equal angles with the base, and terminated by the opposite sides, their sum is the same whatever point be taken.

47. If the point in the last problem is in the base produced, the sum is changed into difference.

48. To divide a given straight line into any number of equal parts.

XXXVI.

49. To bisect a parallelogram by a straight line drawn through a given point in one of its sides.

XXXVII.

50. Of all equal triangles standing upon the same base, the isosceles triangle has the least perimeter.

51. To construct a triangle which shall be equal to a given trapezium, and shall have one side equal to a side of the trapezium.

XXXVIII.

52. Two triangles whose common base is any line taken in the diagonal of a parallelogram, or the diagonal produced, and whose vertices are the opposite angular points of the parallelogram, are equal to one another.

53. Of the three triangles whose common vertex is any point within a parallelogram, and whose bases are two adjacent sides and the included diagonal of the parallelogram, the last is equal to the difference between the other two.

54. The straight line drawn from the vertex of a triangle to the point of bisection of the opposite side, bisects every straight line which is parallel to that side, and terminated by the other sides of the triangle.

55. The straight lines drawn from the three angles of a triangle to the points of bisection of the opposite sides meet all in one. point, which is the point of trisection of each of them; and they divide the triangle into six equal parts.

56. Through a given point lying between two given straight lines, to draw a straight line such that if the three lines are produced they shall all meet in the same point.

57. To bisect a triangle by a straight line drawn through a given point in one of its sides.

XXXIX.

58. If from two points in a straight line equal lines be drawn, making equal angles with it, the line which joins the extremities of these two lines is parallel to the given line.

XL.

59. The two triangles whose vertex is any point within a parallelogram, and whose bases are two of the opposite sides of the parallelogram, are together equal to half the parallelogram. 60. The same is true if the point is taken without the parallelogram, provided it lies between the two bases produced; if otherwise, the difference of the triangles is equal to half the area of the parallelogram.

*This point is the centre of gravity of the triangle.

+