But since every rectangle is double of a triangle of the same base and altitude, (I. 41.)

therefore the rectangle AG, BC, is equal to the three rectangles AB, PD; AC, PF and BC, PE.

Whence the line AG is equal to the sum of the lines PD, PE, PF. If the point P fall on one side of the triangle, or coincide with E: then the triangle ABC' is equal to the two triangles APC, BPA: whence AG is equal to the sum of the two perpendiculars PD, PF. If the point P fall without the base BC of the triangle:

then the triangle ABC is equal to the difference between the sum of the two triangles APC, BPA, and the triangle PCB.

Whence AG is equal to the difference between the sum of PD, PF, and PE.

I.

6. If the straight line AB be divided into two unequal parts in D, and into two unequal parts in E, the rectangle contained by AE, EB, will be greater or less than the rectangle contained by AD, DB, according as E is nearer to, or further from, the middle point of AB, than D.

7. Produce a given straight line in such a manner that the square on the whole line thus produced, shall be equal to twice the square on the given line.

8. If AB be the line so divided in the points C and D, (fig. Euc. II. 5.) shew that AB2 = 4. CD2 +4.AD.DB.

9. Divide a straight line into two parts, such that the sum of their squares may be the least possible.

10. Divide a line into two parts, such that the sum of their squares shall be double the square on another line.

11. Shew that the difference between the squares on the two unequal parts (fig. Euc. II. 9.) is equal to twice the rectangle contained by the whole line, and the part between the points of section.

12. Shew how in all the possible cases, a straight line may be geometrically divided into two such parts, that the sum of their squares shall be equal to a given square.

13. Divide a given straight line into two parts, such that the squares on the whole line and on one of the parts shall be equal to twice the square on the other part.

14. Any rectangle is the half of the rectangle contained by the diameters of the squares on its two sides.

15. If a straight line be divided into two equal and into two unequal parts, the squares on the two unequal parts are equal to twice the rectangle contained by the two unequal parts, together with four times the square on the line between the points of section.

16. If the points C, D be equidistant from the extremities of the straight line AB, shew that the squares constructed on AD and AC, exceed twice the rectangle AC, AD by the square constructed on CD.

17. If any point be taken in the plane of a parallelogram from which perpendiculars are let fall on the diagonal, and on the sides which include it, the rectangle of the diagonal and the perpendicular

on it, is equal to the sum or difference of the rectangles of the sides and the perpendiculars on them.

18. ABCD is a rectangular parallelogram, of which A, C are opposite angles, E any point in BC, Fany point in CD. Prove that twice the area of the triangle AEF together with the rectangle BE, DF is equal to the parallelogram AC.

II.

19. Shew how to produce a given line, so that the rectangle contained by the whole line thus produced, and the produced part, shall be equal to the square (1) on the given line (2) on the part produced.

20. If in the figure Euc. II. 11, we join BF and CH, and produce CH to meet BF in L, CL is perpendicular to BF.

21. If a line be divided, as in Euc. II. 11, the squares on the whole line and one of the parts are together three times the square on the other part.

22. If in the fig. Euc. II. 11, the points F, D be joined cutting AHB, GHK in f, d respectively; then shall Ff= Dd.

III.

23. If from the three angles of a triangle, lines be drawn to the points of bisection of the opposite sides, the squares on the distances between the angles and the common intersection, are together one-third of the squares on the sides of the triangle.

24. ABC is a triangle of which the angle at Cis obtuse, and the angle at B is half a right angle: D is the middle point of AB, and CE is drawn perpendicular to AB. Shew that the square on AC' is double of the squares on AD and DE.

25. If an angle of a triangle be two-thirds of two right angles, shew that the square on the side subtending that angle is equal to the squares on the sides containing it, together with the rectangle contained by those sides.

26. The square described on a straight line drawn from one of the angles at the base of a triangle to the middle point of the opposite side, is equal to the sum or difference of the square on half the side bisected, and the rectangle contained between the base and that part of it, or of it produced, which is intercepted between the same angle and a perpendicular drawn from the vertex.

27. ABC is a triangle of which the angle at Cis obtuse, and the angle at B is half a right angle: D is the middle point of AB, and CE is drawn perpendicular to AB. Shew that the square on AC is double of the squares on AD and DE.

28. Produce one side of a scalene triangle, so that the rectangle under it and the produced part may be equal to the difference of the squares on the other two sides.

29. Given the base of any triangle, the area, and the line bisecting the base, construct the triangle.

IV.

30. Shew that the square on the hypotenuse of a right-angled triangle, is equal to four times the area of the triangle together with the square on the difference of the sides.

31. In the triangle ABC, if AD be the perpendicular let fall upon the side BC; then the square on AC together with the rectangle contained by BC, BD is equal to the square on AB together with the rectangle CB, CD.

32. ABC is a triangle, right angled at C, and CD is the perpendicular let fall from Cupon AB; if HK is equal to the sum of the sides AC, CB, and LM to the sum of AB, CD, shew that the square on HK together with the square on CD is equal to the square on LM.

33. ABC is a triangle having the angle at B a right angle: it is required to find in AB a point P such that the square on AC may exceed the squares on AP and PC by half the square on AB.

34. In a right-angled triangle, the square on that side which is the greater of the two sides containing the right angle, is equal to the rectangle by the sum and difference of the other sides.

35. The hypotenuse AB of a right-angled triangle ABC is trisected in the points D, E; prove that if CD, CE be joined, the sum of the squares on the sides of the triangle CDE is equal to two-thirds of the square on AB.

36. From the hypotenuse of a right-angled triangle portions are cut off equal to the adjacent sides: shew that the square on the middle segment is equivalent to twice the rectangle under the extreme segments.

V.

37. Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base: and conversely.

38. If from one of the equal angles of an isosceles triangle a perpendicular be drawn to the opposite side, the rectangle contained by that side and the segment of it intercepted between the perpendicular and base, is equal to the half of the square described upon the base.

39. If in an isosceles triangle a perpendicular be let fall from one of the equal angles to the opposite side, the square on the perpendicular is equal to the square on the line intercepted between the other equal angle and the perpendicular, together with twice the rectangle contained by the segments of that side.

40. The square on the base of an isosceles triangle whose vertical angle is a right angle, is equal to four times the area of the triangle.

41. Describe an isosceles obtuse-angled triangle, such that the square on the side subtending the obtuse angle may be three times the square on either of the sides containing the obtuse angle.

42. If AB, one of the sides of an isosceles triangle ABC be produced beyond the base to D, so that BD = AB, shew that

CDAB+2.BC2.

43. If ABC be an isosceles triangle, and DE be drawn parallel to the base BC, and EB be joined; prove that BE2 = BC× DE + CE2. 44. If ABC be an isosceles triangle of which the angles at B and Care each double of A; then the square on AC is equal to the square on BC together with the rectangle contained by AC and BC.

VI.

45. Shew that in a parallelogram the squares on the diagonals are equal to the sum of the squares on all the sides.

46. If ABCD be any rectangle, A and C being opposite angles, and O any point either within or without the rectangle:

OA+ OC2 = OB2 + OD2.

47. In any quadrilateral figure, the sum of the squares on the diagonals together with four times the square on the line joining their middle points, is equal to the sum of the squares on all the sides.

48. In any trapezium, if the opposite sides be bisected, the sum of the squares on the other two sides, together with the squares on the diagonals, is equal to the sum of the squares on the bisected sides, together with four times the square on the line joining the points of bisection.

49. The squares on the diagonals of a trapezium are together double the squares on the two lines joining the bisections of the opposite sides.

50. In any trapezium two of whose sides are parallel, the squares on the diagonals are together equal to the squares on its two sides which are not parallel, and twice the rectangle contained by the sides which are parallel.

51. If the two sides of a trapezium be parallel, shew that its area is equal to that of a rectangle contained by its altitude and half the sum of the parallel sides.

52. If a trapezium have two sides parallel, and the other two equal, shew that the rectangle contained by the two parallel sides, together with the square on one of the other sides, will be equal to the square on the straight line joining two opposite angles of the trapezium.

53. If squares be described on the sides of any triangle and the angular points of the squares be joined; the sum of the squares on the sides of the hexagonal figure thus formed is equal to four times the sum of the squares on the sides of the triangle.

VII.

54. Find the side of a square equal to a given equilateral triangle. 55. Find a square which shall be equal to the sum of two given rectilineal figures.

56. To divide a given straight line so that the rectangle under its segments may be equal to a given rectangle.

57. Construct a rectangle equal to a given square and having the difference of its sides equal to a given straight line.

58. Shew how to describe a rectangle equal to a given square, and having one of its sides equal to a given straight line.

BOOK III.

DEFINITIONS.

I. ́

EQUAL circles are those of which the diameters are equal, or from the centers of which the straight lines to the circumferences are equal.

This is not a definition, but a theorem, the truth of which is evident; for, if the circles be applied to one another, so that their centers coincide, the circles must likewise coincide, since the straight lines from the centers are equal.

II.

A straight line is said to touch a circle when it meets the circle, and being produced does not cut it.

III.

Circles are said to touch one another, which meet, but do not cut one another.

IV.

Straight lines are said to be equally distant from the center of a circle, when the perpendiculars drawn to them from the center are equal.

V.

And the straight line on which the greater perpendicular falls, is said to be further from the center.

VI.

A segment of a circle is the figure contained by a straight line, and the arc or the part of the circumference which it cuts off.