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For if it be not, let, if poffible, G be the center, and join GA, Book III. GD, GB. then because DA is equal to DB, and DG common

to the two triangles ADG, BDG, the

two fides AD, DG are equal to the two BD, DG, each to each; and the

bafe GA is equal to the bafe GB, becaufe they are drawn from the center G*. therefore the angle ADG is equal

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c. 8. I.

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angle 4. therefore the angle GDB is a right angle. but FDB is d.ro. Def.z. likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impoffible. therefore G is not the center of the circle ABC. in the fame manner it can be fhewn, that no other point but F is the center; that is, F is the center of the circle ABC. Which was to be found.

COR. From this it is manifeft, that if in a circle a straight line bifect another at right angles, the center of the circle is in the line which bifects the other.

PROP. II. THEO R.

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 ftraight line drawn from A

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*N. B. Whenever the expreffion "straight lines from the center" or " drawn " from the center" occurs, it is 、o be understood that they are drawn to the circumference.

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e. 16. 1.
d. 19. 1.

Book III. AE a fide of the triangle DAE is produced to B, the angle DEB is greater than the angle DAE; but DAE is equal to the angle DBE, therefore the angle DEB is greater than the angle DBE. but to the greater angle the greater fide is oppofited; DB is therefore greater than DE. but DB is equal to DF; wherefore DF is greater than DE, the lefs than the greater, which is impoffible. therefore the ftraight line drawn from A to B does not fall without the circle. in the fame manner, it may be demonftrated that it does not fall upon the circumference. it falls therefore within it. Wherefore if any two points, &c. Q. E. D.

a. 2. 3.

ს. 8. 1.

IF

PROP. III. THEOR.

Fa ftraight line drawn thro' the center of a circle, bifect a ftraight line in it which does not pass thro' the center, it shall cut it at right angles. and if it cuts it at right angles, it fhall bifect it.

Let ABC be a circle; and let CD a straight line drawn thro' the center bifect any ftraight line AB, which does not pafs thro' the center, in the point F. it cuts it alfo at right angles.

a

Take E the center of the circle, and join EA, EB. then becaufe AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two fides in the one equal to two fides in the other. and the bafe EA is equal to the

c

bafe EB; therefore the angle AFE is equal to the angle BFE. but when a straight line ftanding upon another makes the adjacent angles equal to one another, each c.10.Def.1. of them is a right angle. therefore each of the angles AFE, BFE is a right angle; wherefore the straight line CD drawn thro' the center bifecting another AB that does not pass thro' the center, cuts the fame at right angles.

d. 5. 1.

E

D

But let CD cut AB at right angles; CD alfo bifects it, that is, AF is equal to FB.

The fame conftruction being made, becaufe EA, EB from the center 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 an.

gles in one equal to two angles in the other, and the fide EF which BookIII. is oppofite to one of the equal angles in each, is common to both; therefore the other fides are equal, AF therefore is equal to FB.c. 26, 1. Wherefore if a straight line, &c. Q. E. D.

IF

PROP. IV. THEO R.

F in a circle two straight lines cut one another which do not both pass thro' the center, they do not bifect each the other.

Let ABCD be a circle, and AC, BD two ftraight lines in it which cut one another in the point E, and do not both pass thro' the center. AC, BD do not bifect one another.

For, if it is poffible, let AE be equal to EC, and BE to ED. if one of the lines pafs thro' the center, it is plain that it cannot be bifected by the other which does not pass thro' the center. but if neither of them pafs thro' the center, take F the center of the circle, and join EF. and because FE a ftraight line thro' the center, bifects another AC which does A not pass thro' the center, it fhall cut it at right angles; wherefore FEA is a right angle. again, because the straight

b

a. x.

F

B

E

line FE bifects the straight line BD which does not pass thro' the center, it shall cut it at right angles; wherefore FEB is a right angle. and FEA was shewn to be a right angle; therefore FEA is equal to the angle FEB, the less to the greater, which is impoffible. therefore AC, BD do not bifect 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 fame center.

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

b. 3. 3.

Book III.

For, if it be poffible, let E be their center; join EC, and draw any ftraight line EFG meeting

them in F and G. and because E

is the center of the circle ABC,
CE is equal to EF. again, because
E is the center of the circle CDG, A
CE is equal to EG. but CE was
fhewn to be equal to EF; there-
fore EF is equal to EG, the lefs
to the greater, which is impoffible.
therefore E is not the center of

E

B

the circles ABC, CDG. Wherefore if two circles, &c. Q. E. D.

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IF

two circles touch one another internally, they fhall not have the fame center.

Let the two circles ABC, CDE touch one another internally in the point C. they have not the fame center.

For if they can, let it be F; join FC, and draw any straight

line FEB meeting them in E and B.

C

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cles ABC, CDE. Therefore if two circles, &c. Q. E. D.

PROP. VII.

THEOR.

Book III.

IF

F any point be taken in the diameter of a circle, which is not the center, of all the straight lines which can be drawn from it to the circumference, the greateft is that in which the center is, and the other part of that diameter is the leaft; and of any others, that which is nearer to the line which paffes thro' the center is always greater than one more remote. and from the fame point there can be drawn only two straight lines that are equal to one another, one upon each fide 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 center. let the center 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 leaft; and of the others, FB is greater than FC, and FC than FG.

Join BE, CE, GE; and because two fides of a triangle are greater than the third, BE, EF are greater than BF; but AE is a. 10.

2

b

A

B

E

K

b. 24.

D H

equal to EB, therefore AE, EF, that
is AF, is greater than BF. again, be-
caufe BE is equal to CE, and FE
common to the triangles BEF, CEF;
the two fides BE, EF are equal to the
two CE, EF; but the angle BEF is
greater than the angle CEF, therefore
the base BF is greater than the bafe
FC. for the fame reafon, CF is grea-
ter than GF. again, because GF, FE
are greater than EG, and EG is equal to ED; 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 circumference;
and BF is greater than CF, and CF than GF.

Also there can be drawn only two equal ftraight lines from the point F to the circumference, one upon each fide of the shortest line

E

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