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ELEMENTS

or

GEOMETRY.

BOOK III.

DEFINITIONS.

A. The radius of a circle is the straight line drawn from the centre to the circumference.

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

And that line which has but one point in common with the circumference, is called a tangent, and the point in common, the point of contact.

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

3. 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.

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

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the centre.

B. Any portion of the circumference is called an arc.

The chord or subtense of an arc is the straight line which joins its two extremities.

C. A straight line is said to be inscribed in a circle, when the extremities of it are in the circumference of the circle. And any straight line which meets the circle in two points, is called a secant.

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

6. 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.

An inscribed triangle, is one which has its three angular points in the circumference.

And, generally, an inscribed figure is one, of which all the angles are in the circumference. The circle is said to circumscribe such a figure.

7. And an angle is said to insist or stand upon the arc intercepted between the straight lines which contain the angle.

This is usually called an angle at the centre. The

angles at the circumference and centre, are
both said to be subtended by the chords or
arcs which their sides include.

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

9. Similar segments of a circle, are those in which the angles are equal, or which contain equal angles.

PROP. I. PROB.

To find the centre of a given circle.

Let ABC be the given circle; it is required to find its centre. Draw within it any straight line AB, and bisect (10. 1.) it in D; from the point D draw (11. 1.) DC at right angles to AB, and produce it to E, and bisect CE in F: the point F is the centre of the cirele ABC

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For, if it be not, let, if possible, G be the centre, and join GA, GD, GB: Then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG are equal to the two BD, DG, each to each; and the base GA is equal to the base GB, because they are radii of the same circle: therefore the angle ADG is equal (8. 1.) to the angle GDB: But when a straight line standing upon another straight line makes the adjacent angles equal to one another, each of the angles is a right angle (7. def. 1.) Therefore the angle GDB is a right angle: But FDB is likewise a right angle: wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impos

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sible Therefore G is not the centre of the circle ABC: In the same manner it can be shown that no other point but F is the centre: that is F is the centre of the circle ABC.

COR. 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 line which bisects the other.

PROP. II. THEOR.

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, B any two points in the circumference ; he straight line drawn from A to B shall fall within the circle.

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D

E

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Take any point in AB as E; find D (1. 3.) he centre of the circle ABC; join AD, DB and DE, and let DE meet the circumference in F. Then, because DA is equal to DB, the angle DAB is equal (5. 1.) to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater (16. 1.) than the angle DAE; but DAE is equal to the angle DBE; therefore the angle DEB is greater than the angle DBE: Now to the greater angle the greater side is opposite (19. 1.); DB is therefore greater than DE: but BD is equal to DF; wherefore DF is greater than DE, and the point E is therefore within the circle. The same may be demonstrated of any other point between A and B, therefore AB is within the circle.

COR. Every point, moreover, in the production of AB, is farther from the centre than the circumference.

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If a straight line drawn through the centre of a circle bisect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles; and if it cut it at right angles, it will bisect it.

Let ABC be a circle, and let CD, a straight line drawn through the centre, bisect any straight line AB, which does not pass through the centre, in the point F; it cuts it also at right angles.

And

Take (1. 3.) E the centre of the circle, and join EA, EB. Then because AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two sides in the one equal to two sides in the other: but the base EA is equal to the base EB; therefore the angle AFE is equal (8. 1.) to the angle BFE. when a straight line standing upon another makes the adjacent angles equal to one another, each of them is a right (7. Def. 1.) angle: Therefore each of the angles AFE, BFE is a right angle; wherefore the straight line CD, drawn through the centre

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bisecting AB, which does not pass through the centre, cuts AB at righ angles.

Again, let CD cut AB at right angles; CD also bisects AB, that is, AF is equal to FB.

The same construction being made, because the radii EA, FB are equal to one another, the angle EAF is equal (5. 1.) to the angle EBF; and the right angle AFE is equal to the right angle BFE: Therefore, in the two triangles EAF, EBF, there are two angles in one equal to two angles in the other; now the side EF, which is opposite to one of the equal angles in each, is common to both; therefore the other sides are equal to (26. 1.): AF therefore is equal to FB.

COR. 1. Hence, the perpendicular through the middle of a chord, passes through the centre; for this perpendicular is the same as the one let fall from the centre on the same chord, since both of them passes through the middle of the chord.

It likewise follows, that the perpendicular, drawn through the middle of a chord, and terminated both ways by the circumference of the circle, is a diameter, and the middle point of that diameter is therefore the centre of the circle.

PROP. IV. THEOR.

If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other.

Let ABCD be a circle, and AC, BD two straight lines in it, which cut one another in the point E, and do not both pass through the centre: AC, BD do not bisect one another.

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E

B

For if it is possible, let AE be equal to EC, and BE to ED; if one of the lines pass through the centre, it is plain that it cannot be bisected by the other, which does not pass through the centre. But if neither of them pass through the centre, take (1. 3.) F the centre A of the circle, and join EF: and because FE, a straight line through the centre, bisects another AC, which does not pass through the centre, it must cut it at right (3. 3.) angles; wherefore FEA is a right angle. Again, because the straight line FE bisects the straight line BD, which does not pass through the centre, it must cut it at right (3. 3.) angles; wherefore FEB is a right angle and FEA was shown to be a right angle: therefore FEA is equal to the angle FEB, the less to the greater, which is impossible; therefore AC, BD, do not bisect one another.

PROP. V. THEOR.

If two circles cut one another, they cannot have the same centre.

Let the two circles ABC, CDG cut one another in the points B, C ; they have not the same centre.

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For, if it be possible, let E be their centre: join EC, and draw any straight line EFG meeting the circles in F and G: and because E is the centre of the circle ABC, CE is equal to EF: Again, because E is the centre of the circle CDG, CE is equal to EG: but CE was shown to be equal to EF, therefore EF is equal to EG, the less to the greater, which is impossible: therefore E is not the centre of the circles, ABC, CDG.

PROP. VI. THEOR.

If two circles touch one another internally, they cannot have the same centre

Let the two circles ABC, CDE, touch one another internally in the point C; they have not the same centre.

For, if they have, let it be F; join FC, and draw any straight line FEB meeting the circles in E and B; and because F is the centre of the circle ABC, CF is equal to FB; also, be cause F is the centre of the circle CDE, CF is equal to FE: but CF was shown to be equal to FB; therefore FE is equal to FB, the less to the greater, which is impossible: Wherefore F is not the centre of the circles ABC, CDE.

PROP. VII. THEOR.

C

F

E

B

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If any point be taken in the diameter of a circle which is not the centre, of ail the straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is, and the other part of that diameter is the least; and, of any others, that which is nearer to the line passing through the centre is always greater than one more remote from it; And from the same point there can be drawn only two straight lines that are equal to one another, one upon each side of the shortest line.

Let ABCD be a circle, and AD its diameter, in which let any point f be taken which is not the centre: let the centre be E; of all the straight lines FB, FC, FG, &c. that can be drawn from F to the circumference, FA is the greatest; and FD, the other part of the diameter AD, is the least; and of the others, FB is greater than FC, and FC than FG.

Join BE, CE, GE; and because two sides of a triangle are greate (20. 1.) than the third, BE, EF are greater than BF; but AE is equal to EB; therefore AE and EF, that is, AF, is greater than BF: Again, be cause BE is equal to CE, and FE common to the triangles BEF, CEF.

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