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But if they are not equal, produce one of them BE to F, make EF equal

to ED,

[I. 3.

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H

circle BHF, and produce DEto H.

The square described on EH shall be equal to the given rectilineal figure A.

Join GH. Then, because the straight line BF is divided into two equal parts at the point G, and into two unequal parts at the point E, the rectangle BE, EF, together with the square on GE, is equal to the square on GF.

But GF is equal to GH.

[II. 5.

Therefore the rectangle BE, EF, together with the square on GE, is equal to the square on GH.

But the square on GH is equal to the squares on GE, EH;[1.47. therefore the rectangle BE, EF, together with the square on GE, is equal to the squares on GE, EH.

Take away the square on GE, which is common to both; therefore the rectangle BE, EF is equal to the square on EH.

[Axiom 3.

But the rectangle contained by BE, EF is the parallelogram BD,

because EF is equal to ED.

[Construction.

[Construction.

Therefore BD is equal to the square on EH.
But BD is equal to the rectilineal figure A.
Therefore the square on EH is equal to the rectilineal
figure A.

Wherefore a square has been made equal to the given rectilineal figure A, namely, the square described on EH. Q.E.F.

BOOK III.

DEFINITIONS.

1. EQUAL circles are those of which the diameters are equal, or from the centres 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 centres coincide, the circles must likewise coincide, since the straight lines from the centres are equal.

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

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

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

5. And the straight line on which the greater perpendicular falls, is said to be farther from the centre.

6. A segment of a circle is the figure contained by a straight line and the circumference it cuts off.

7. The angle of a segment is that which is contained by the straight line and the circumference.

8. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment to the extremities of the straight line which is the base of the segment.

9. And an angle is said to insist or stand on the circumference intercepted between the straight lines which contain the angle.

10. A sector of a circle is the figure contained by two straight lines drawn from the centre, and the circumference between them.

11. Similar segments of circles are those in which the angles are equal, or which contain equal angles.

[Note. In the following propositions, whenever the expression "straight lines from the centre," or "drawn from the centre," occurs, it is to be understood that the lines are drawn to the circumference.

Any portion of the circumference is called an arc.]

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To find the centre of a given circle.

Let ABC be the given circle: it is required to find its centre.

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The point F shall be the centre of the circle ABC.

For if F be not the centre,

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if possible, let G be the centre; and join GA, GD, GB. Then, because DA is equal to DB,

[Construction. and DG is common to the two triangles ADG, BDG ; the two sides AD, DG are equal to the two sides BD, DG, each to each;

and the base GA is equal to the base GB, because they are drawn from the centre G; [I. Definition 15.

therefore the angle ADG is equal to the angle BDG. [I. 8. But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle; [I. Definition 10.

therefore the angle BDG is a right angle. But the angle BDF is also a right angle.

[Construction.

Therefore the angle BDG is equal to the angle BDF, [Ax. 11. the less to the greater; which is impossible.

Therefore G is not the centre of the circle ABC.

In the same manner it may be shewn that no other point out of the line CE is the centre;

and since CE is bisected at F, any other point in CE divides it into unequal parts, and cannot be the centre. Therefore no point but F is the centre ;

that is, Fis the centre of the circle ABC: which was to be found.

COROLLARY. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the straight line which bisects the other.

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If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

Let ABC be a circle, and A and B any two points in the circumference: the straight line drawn from A to B shall fall within the circle.

For if it do not, let it fall, if possible, without, as AEB. Find D the centre of the circle ABC; [III. 1.

and join DA, DB; in the arc AB take any point F, join DF, and produce it to meet the straight line AB at E.

Then, because DA is equal to DB,

[I. Definition 15.

A

E B

the angle DAB is equal to the angle DBA.

[I. 5. And because AE, a side of the triangle DAE, is produced to B, the exterior angle DEB is greater than the interior opposite angle DAE.

[I. 16. But the angle DAE was shewn to be equal to the angle DBE; therefore the angle DEB is greater than the angle DBE. But the greater angle is subtended by the greater side; [I. 19. therefore DB is greater than DE.

But DB is equal to DF;

[I. Definition 15. therefore DF is greater than DE, the less than the greater; which is impossible.

Therefore the straight line drawn from A to B does not fall without the circle.

In the same manner it may be shewn that it does not fall on the circumference.

Therefore it falls within the circle.

Wherefore, if any two points &c. Q.E.D.

PROPOSITION 3. THEOREM.

If a straight line drawn through the centre of a circle, bisect a straight line in it which does not pass through the

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