join AF;

then shall AFDE be equal to the figure ABCDE.

Because the triangles AFC and ABC are on the same base AC

and of equal altitude,

therefore the triangle AFC is equal to the triangle ABC.

To each add the figure ACDE,

then the figure AFDE is equal to the figure ABCDE.

II. 2, Cor. 1.

In like manner if EF be joined, and AG be drawn parallel to EF to meet DF produced at G, the triangle EGD will be equal to the figure AFDE, and therefore to the figure ABCDE.

Q.E.F.

Ex. 28. ABC is a triangle, D any point in BC produced, find a point E in AB such that the triangle EBD may be equal to ABC.

Ex. 29. Bisect a triangle by a straight line drawn through a given point in one of the sides.

PROB. 6. To divide a given straight line, either internally or externally, into two segments such that the rectangle contained by the given line and one of the segments may be equal to the square on the other segment.

Let AB be the given straight line:

it is required to divide it internally and externally so that the rectangle contained by AB and one segment may be equal to the square on the other segment.

join EA, and with centre E and radius EA draw a circle cutting BC produced in F and F';

on BF and BF' draw the squares BFHG and BF'H'G':

then shall the rectangle contained by AB and AG be equal to the square on BG,

and the rectangle contained by AB and AG' to the square on

BG'.

Complete the rectangles FHKC and F'H'K'C.

Then the square on AB is equal to the difference of the squares on AE and BE; II. 9.

but the difference of the squares on AE and BE is equal to the rectangle contained by CF, their sum, and BF their difference, i.e., to the figure FK, and, in like manner, to the figure F'K';

II. 8.

therefore the figure AC, which is the square on AB, is equal to each of the figures FK and F'K'.

Therefore the remainder AK, which is the rectangle contained by AB and AG, is equal to the remainder FG, which is the square on BG; and the whole AK', which is the rectangle contained by AB and AG', is equal to the whole F'G' which is the square on BG'.

Q.E.F.

Ex. 30. Supposing AB to be a foot long, shew that the lengths of BG and BG' in inches are the roots of the equation x2+12x = 144, and hence calculate the lengths of BG and BG' correctly to a tenth of an inch.

EXERCISES ON BOOK II.

31. Two equal triangles are on the same base and on opposite sides of it: shew that the base bisects the straight line joining their vertices.

32. The diagonals of a quadrilateral intersect at right angles : shew that the rectangle contained by the diagonals is double the quadrilateral.

33. The diagonals of a quadrilateral intersect at right angles:

shew that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other pair. *34. ABC is an equilateral triangle and AD is perpendicular to BC: shew that the square on AD is equal to three times the square on BD.

35. Prove that the rectangle contained by the segments of a given straight line is greatest when the segments are equal. 36. If a straight line is divided into two parts, prove that the sum of the squares on the parts is least when the line is bisected.

37. A triangle whose base is one of the non-parallel sides of a trapezium, and vertex the middle point of the opposite side, is equal to half the trapezium.

38. D is the middle point of the base BC of the triangle ABC, BE bisects AD and meets AC at E: shew that the triangle BEC is double the triangle ABE.

*39. The difference of the squares on two sides of a triangle is

equal to the difference of the squares on the segments of

*

the base made by a perpendicular from the opposite

vertex.

40. If O be a point in the base BC, or BC produced, of the isosceles triangle ABC, the difference of the squares on OA and AB is equal to the rectangle contained by OB and OC. 41. If from the middle point of one of the sides of a right

angled triangle a perpendicular be drawn to the hypotenuse, the difference of the squares on the segments of the hypotenuse is equal to the square on the remaining side. 42. Prove that three times the sum of the squares on the sides of a triangle is equal to four times the sum of the squares on the lines drawn from the vertices to the middle points of the opposite sides.

43. The hypotenuses of three isosceles right-angled triangles form a right-angled triangle: shew that one of the isosceles triangles is equal to the sum of the other two.

44. In any quadrilateral the sum of the squares on the diagonals is double the sum of the squares on the lines joining the middle points of the opposite sides.

45. Prove, by constructions similar to that in the First Proof of

Theor. 9, that in a triangle the sum of the squares on the sides containing an acute angle is greater, and on those containing an obtuse angle less, than the square on the other side.

46. Construct a square equal to the difference of two given

squares.

47. Divide a given straight line into two segments such that

the sum of their squares may be equal to the square on another given straight line. When must the straight line be divided externally?