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See N.

* 10. 1.

11. 1.

+ Constr.

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* it in D; from the point D draw* DC at right angles to AB, and produce it to E, and bisect CE in F: the point F shall be the centre of the circle ABC.

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, DĞ, are equal to the two BD, DG, each to each; and the base GA is † 15 Def. 1. equal to the base GB, because they

*8.1.

*10 Def. 1. + Constr.

† 1 Ax.

FG

ID

E

B

are drawn from the centre G+: there-
fore the angle ADG is equal 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; therefore the
angle GDB is a right angle: but FDB is likewise at
right angle; wherefore the angle FDB is equal to
the angle GDB, the greater to the less, which is im-
possible: therefore G is not the centre of the circle
ABC. In the same manner it can be shewn that no
other point but F is the centre; that is, F is the centre
of the circle ABC. Which was to be found.

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; the straight line drawn from A to B shall fall within the circle.

N. B.-Whenever the expression "straight lines from the centre," or drawn" from the centre" occurs, it is to be understood that they are drawn to the circumference.

For if it do not, let it fall, if possible, without, as AEB: find* D the centre of the circle ABC; and join DA, DB; in the circumference AB take any point F, join DF, and produce it to E: then because DA is equal+ to DB, the angle DAB is equal*

[graphic]

D

1. 3.

†15 Def. 1.

5. 1.

to the angle DBA: and because AE, a side of the tri-
angle DAE, is produced to B, the exterior angle DEB
is greater than the angle DAE: but DAE was proved 16. 1.
equal to the angle DBE; therefore the angle DEB is
greater than the angle DBE: but to the greater angle
the greater side is* opposite, therefore DB is greater *19. 1.
than DE: but DB is equal+ to DF; wherefore DF is †15 Def. 1.
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 man-
ner, it may be demonstrated that it does not fall upon
the circumference; therefore it falls within it. Where-
fore, if any two points, &c. Q. E. D.

PROP. III: THEOR.

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

Take E the centre of the circle, and join EA, EB. 1.3. Then, because AF is equal to FB, and FE common † Hyp. to the two triangles AFE, BFE, there are two sides in the one equal to two sides in the other, each to each; and the base EA is equal + to the base EB; therefore the angle AFE is equal to the angle BFE: but when a straight line standing upon another straight line makes the adjacent angles equal to one another, each of them is a

† 15 Def. 1.

*8.1.

B

right angle; therefore each of the angles AFE, BFE, 10 Def. 1.
is a right angle: wherefore the straight line CD, drawn
through the centre, bisecting another AB that does not
pass through the centre, cuts the same at right angles.

B

[graphic]

* 5. 1.

†10 Def.1.

But let CD cut AB at right angles; CD shall also bisect it, that is, AF shall be equal to FB.

**

The same construction being made, because EA, EB, † 15 Def. 1. from the centre are equal to one another, the angle EAF is equal 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 the one equal to two angles in the other, each to each; and the side EF, which is opposite to one of the equal angles in each, is common to both; therefore the other sides are * equal; therefore AF is equal to FB. Wherefore, if a straight line, &c. Q. E. D.

26. 1.

* 1. 3.

+ Hyp.

*3.3.

+ Hyp.

* 3. 3. † 1 Ax.

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 shall not bisect one another.

A

F

B

For, if it be 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* F the centre 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 cuts it at right 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 cuts it at right angles: wherefore FEB is a right angle: but FEA was shewn 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. Wherefore, if in a circle, &c. Q. E. D.

*

PROP. V. THEOR.

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

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

For, if it be possible, let E be their centre: join EC, and draw any straight line EFG meeting them in F and G: and because E is the centre of the circle ABC, EC is equal to EF: again, because E is the centre of the circle CDG, EC is equal + to EG: but EC was shewn to be equal to EF; therefore EF is equal to EG, the less to the greater, which is impossible. Therefore

[graphic]

D

+15 Def. 1.

† 15 Def. 1.

E

† 1 Ax.

E is not the centre of the circles ABC, CDG. Wherefore, if two circles, &c. Q. E. D.

PROP. VI. THEOR.

If two circles touch one another internally, they shall not have the same centre.

Let the two circles ABC, CDE, touch one another internally in the point C: they shall not have the same

centre.

F

any

† 15 Def. 1.

B

† 15 Def. 1.

For, if they have, let it be F: join FC and draw straight line FEB, meeting them in E and B and because F is the centre of the circle ABC, FC is equal+ to FB; also, because F is the centre of the circle CDE, FC is equal to FE: but FC was shewn to be equal to FB; therefore FE is equal to FB, the less to the greater, which is impossible: therefore F is not the centre of the circles ABC, CDE. Therefore, if two circles, &c. Q. E. D.

[graphic]

D

† 1 Ax.

[blocks in formation]

If any point be taken in the diameter of a circle which is not the centre, of all 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 which passes through the centre is always greater than one more remote: 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.

* 20.1.

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 shall be the greatest, and FD, the other part of the diameter AD, shall be the least: and of the others, FB shall be greater than FC, and FC greater than FG.

Join BE, CE, GE: and because two sides of a triangle are greater* than the third, therefore BE, EF, are +15 Def. 1. greater than BF: but AE is equal+ to

+9 Ax. 1.

* 24. 1.

*20.1.

† 5 Ax.

23.1.

+ Constr.

*

BE; therefore AE, EF, that is AF, is
greater than BF: again, because BE is
equal to CE, and FE common to the tri-
angles BEF, CEF, the two sides BE,
EF, are equal to the two CE, EF, each
to each; but the angle BEF is greater+
than the angle CEF; therefore the base
BF is greater than the base CF: for the same reason,
CF is greater than GF: again, because GF, FE are
greater than EG, and EG is equal to ED; therefore
GF, FE are greater than ED: take away the common
part FE, and the remainder GF is greater+ than the
remainder FD. Therefore FA is the greatest, and FD-
the least of all the straight lines from F to the circum-
ference; and BF is greater than CF, and CF than GF.

*

Also there can be drawn only two equal straight lines from the point F to the circumference, one upon each side of the shortest line FD. At the point E in the straight line EF, make* the angle FEH equal to the +15 Def. 1. angle FEG, and join FH: then because GE is equal + to EH, and EF common to the two triangles GEF, HEF; the two sides GE, EF are equal to the two HE, EF, each to each; and the angle GEF is equal + to the angle HEF; therefore the base FG is equal to the base FH: but, besides FH, no other straight line can be drawn from F to the circumference equal to FG: for, if there can, let it be FK: and because FK is equal to FG, and FG to FH, FK is equal to FH; that is, a line nearer to that which passes through the centre, is equal to one which is more remote; which has been proved to be impossible. Therefore, if any point be taken, &c. Q. E. D.

* 4.1.

† 1 Ax.

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