In the triangles DAF, EAF,

AD is equal to AE, and AF common,

(Constr.)

therefore the two sides DA, AF are equal to the two EA, AF,

each to each;

also, the base DF is equal to the base EF.

(Constr.) Therefore the angle DAF is equal to the angle EAF; (I. 8) that is, the angle BAC is bisected by the straight line AF. Q. E. F.

EXERCISE XIV.

1. Apply Prop. VIII. to the figure of IX. so as to prove that the angles ADF, AEF are equal.

2. If a quadrilateral have two adjacent sides equal to one another, and also the other two sides equal, prove that one of its diagonals bisects the angles through which it passes.

3. Then prove that this diagonal bisects the other diagonal.

4. If a quadrilateral have the two sides which contain one angle equal to one another, and a diagonal bisecting the same angle, prove that the remaining sides of the quadrilateral will be equal to one another.

41. PROPOSITION X.-PROBLEM.

To bisect a given finite straight line.

Let AB be the given straight line.

It is required to divide AB into two equal parts. Upon AB describe the equilateral triangle ABC; and bisect the angle ACB by the straight line CD,

(I. 1) (I. 9)

Because, in the two triangles ACD, BCD,

AC is equal to CB, and CD common;

(Constr.)

the two sides AC, CD are equal to BC, CD, each to each;

and the angle ACD is equal to BCD.

(Constr.)

Therefore the base AD is equal to the base DB; wherefore AB is bisected in the point D.

EXERCISE XV.

(I. 4)

Q. E. F.

1. Supposing, with ruler and compass, you perform the whole construction described; how many equilateral triangles would you employ?

2. Which of Euclid's previous figures would be appropriate?

3. Divide a given finite straight line into four equal parts.

4. In the proof of Prop. X. we have drawn one conclusion; proceed to draw three others.

5. Which of these conclusions was known before; and which shows that CD has been drawn at right angles to AB? Quote the appropriate definition.

6. In the figure to I. 10 draw BE bisecting the angle B and meeting CA in E; then show that AD and CE are equal to one another.

7. Join DE in the same construction, and prove that ADE is isosceles. 8. Also that BE and CD are equal to one another in the triangles BDC, BEC.

9. Suppose BE and CD to meet in H, in the figure just made; then prove that HBC is an isosceles triangle.

42. PROPOSITION XI.-PROBLEM.

To draw a straight line at right angles to a given straight line from a given point in the same.

Let AB be the given straight line, and C the given point. It is required to draw from C a straight line at right angles to AB.

Now, in the triangles DCF, ECF,

DC is equal to EC, and CF is common;

(Constr.)

therefore the two sides DC, CF are equal to the two EC, CF,

each to each,

and the base DF is equal to the base EF.

(Constr.)

Therefore the angle DCF is equal to the angle ECF; (1.8) and these are two adjacent angles;

therefore each of the angles DCF, ECF is a right angle.

(Def. 10) Wherefore, from the given point C, in the given straight line AB, CF has been drawn at right angles to AB.

EXERCISE XVI.

Q. E. F.

Def.-A chord of a circle is a straight line joining any two points in its circumference.

1. Prove the proposition; using the letters L, M, N, P, Q, R, and making the equilateral triangle below the given line.

2. If AB were finite, and the given point at one extremity, how would you add to the construction given without further altering the proposition ?

3. Given a finite straight line, show how to draw a straight line both bisecting it and at right angles to it. Say what Proposition has used the required construction already.

4. Show that the construction of I. 1, with the addition of a certain line, will answer the same purpose.

5. In the figure of I. 11, show that any point in CF is equidistant from D and E.

6. Prove that a straight line drawn at right angles to the base of an isosceles triangle from its middle point will pass through the vertex.

7. Make a right-angled triangle which shall also be isosceles.

8. Construct a quadrilateral which shall have at least two of its angles right angles.

9. In the figure of Proposition XI., suppose F taken to be the centre of a circle, and FD its radius; show that this circle will cut AB in E.

10. Take any circle; draw a chord joining any two points in its circumference; bisect this chord; join the point of bisection to the centre of the circle. Show that the line last drawn is at right angles to the chord. (Join the centre to the ends of the chord, and prove after the manner of Proposition XI.)

E

43. PROPOSITION XII.-PROBLEM.

To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.

Let AB be the given straight line, which may be produced any length both ways, and let C be a point without it.

It is required to draw from C a perpendicular to AB.
Take any point D upon the other side of AB;

with centre C, at the distance CD, describe the circle EFG, meeting AB in F and G;

bisect FG in H,

and join CH.

Then CH shall be drawn perpendicular to AB.
Join CF and CG.

(Post. 3) (I. 10)

therefore the two sides FH, HC are equal to the two GH, HC,

each to each;

and the base CF is equal to the base CG.

Therefore the angle FHC is equal to GHC;

(Def. 15)

(I. 8)

and these are adjacent angles.

Hence each of the angles FHC, GHC is a right angle, and the straight line CH is perpendicular to AB. (Def. 10) Wherefore, from the given point C, a perpendicular CH

has been drawn to the given straight line AB.

EXERCISE XVII.-(RECAPITULATORY.)

Q. E. F.

1. ABCD is a circle, whose centre is O, and the chords AB, BC, CD are respectively 3 ft., 4 ft., 3 ft. long; prove that two of the angles A0B, BOC, COD are equal.

2. Deduce a method of testing the equality of two angles at one point. 3. Make a straight line equal to one-and-a-half times a given straight line.

4. At a given point in a given straight line, construct an angle equal to a right angle.

5. At a given point in a given straight line, construct an angle equal to half a right angle.

6. Make a right-angled triangle, having each of the sides containing the right angle equal to a given straight line.

7. Construct a right-angled triangle, having one of the sides containing the right angle equal to a given straight line, and the other half as long.

44. PARTS OF A PROPOSITION.

We have already referred to the enunciation, construction, and demonstration of a proposition. There is but little to add in order to complete, sufficiently for us, the analysis of a proposition.

The student should notice that each enunciation printed in italics is followed by a re-statement of itself referred by means of letters to a specific figure. These two statements are distinguished from each other by calling the former the general enunciation, and the other the particular enunciation; that is, the enunciation which applies to the particular diagram which we find it convenient to construct for the purposes of the proof. Thus, in a proposition, we have as its leading divisions :—

45. MUTUAL RELATIONS OF PROPOSITIONS VII.-XII.

Of these, the leading proposition is VIII.; which is the second case of equality, in all respects, of two triangles under given conditions.

Proposition VII. is accessory to it, and is not elsewhere applied in Euclid's Elements; while IX.-XII. are problems depending upon it for their solution.

EXAMINATION XIX.

1. Describe the relation in which Prop. VII. stands to Prop. VIII. 2. Describe the relation between Prop. VIII. and the four problems which follow it. 3. Show that Prop. VIII. is applied in all these four problems. 4. Contrast the relation of problems to theorems in Props. I.—VI. with the relation of problems to theorems in Props. VII.—XII. 5. Distinguish