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C. 23. 1.

Book III. FD. at the point E in the straight line EF, make the angle FEH equal to the angle GEF, and join FH.

A
then because GE is equal to EH, and B
EF common to the two triangles GEF,

C с
HEF; the two sides GE, EF are equal
to the two HE, EF; and the angle

E
GEFisequal to the angle HEF, there-
d. 4. I.

fore the base FG is equal d to the
base FH. but besides FH no other
straight line can be drawn from F to

D
the circumference equal to FG. for

H 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 thro' the center is equal to one which is more remote ; which is impossible. Therefore if any point be taken, &c. Q. E. D.

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PRO P. VIII. THEOR.
F any point be taken without a circle, and straight

lines be drawn from it to the circumference, whereof one passes thro' the center; of those which fall upon the concave circumference the greatest is that which passes thro' the center; and of the rest, that which is nearer to that thro’ the center is always greater than the more remote. but of those which fall upon the convex circumference, the least is that be tween the point without the circle, and the diameter ; and of the rest, that which is nearer to the least is always less than the more remote. and only two equal straight lines can be drawn from the point unto the circumference, one upon each side of the leaft.

Let ABC be a circle, and D any point without it, from which let the straight lines DA, DE, DF, DC be drawn to the circumference, whereof DA passes thro' the center of those which fall upon the concave part of the circumference AEFC, the greatest is AD which passes thro' the center; and the nearer to it is always greater than the more remote, viz. DE than DF, and DF than DC. but of those which fall upon the convex circumference HLKG, the

least is DG between the point D and the diameter AG; and the Book III. nearer to it is always less than the more remote, viz. DK than DL, and DL than DH.

Take M the center of the circle ABC, and join ME, MF, MC, 6. 1. 3. MK, ML, MH. and because AM is equal to ME, add MD to each, therefore AD is equal to EM, MD; but EM, MD are greater b. 20. fi than ED, therefore also AD is greater than ED. again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD are equal to FM,

D MD; but the angle EMD is greater than the angle FMD, therefore the base ED is greater than the base

A 24.1. FD. in like manner it may

be shewn that FD is greater than CD. therefore DA is the greatest ; and DE

GIB

N greater than DF, and DF than DC.

Н. änd because MK, KD are greater 6 than MD. and MK is equal to MG, the remainder KD is greater than

M

d. 4. Az the remainder GD, that is, GD is less than KD. and because MK,DK áre drawn to the point K within the F triangle MLD from M, the extremities of its side MD; MK, KD E

A are less than ML, LD, whereof

6. 21. I MK is equal to ML, therefore the remainder DK is lefs than the femainder DL. in like manner it may be shewn that DL is less than DH. therefore DG is the leaft, and DK less than DL, and DL than DH. Also there can be drawn only two equal straight lines from the point D to the circumference, one upon each side of the least. at the point M in the straight line MD, make the angle DMB equal to the angle DMK, and join DB. and because MK is equal to MB, and MD common to the triangles KMD, BMD, the two sides KM, MD are equal to the two BM, MD; and the angle KMD is equal to the angle BMD, therefore the base DK is equal f to the f. 4.si bafe DB. but besides DB there can be no straight line drawn from D to the circumference equal to DK. for if there can, let it be DN; and because DK is equal to DN, and also to DB, therefore DB is equal to DN, that is the nearer to the least equal to the more remote, which is impossible. If therefore any point, &c. Q. E. D.

Ε Ι Ε

-L

Book III.

PRO P. IX. THEOR.
IT:
F a point be taken within a circle, from which there

fall more than two equal straight lines to the circumference, that point is the center of the circle.

Let the point D be taken within the circle ABC, from which to the circumference there fall more than two equal straight lines, viz. DA, DB, DC. the point D is the center of the circle.

For if not, let E be the center, join DE and produce it to the circumference in F, G; then FG is a diameter of the circle ABC. and be

DE cause in FG the diameter of the F

G circle ABC there is taken the point D which is not the center, DG shall be the greatest line from it to the circumference, and DC greater than

A B DB, and DB than DA. but they are likewise equal, which is impossible. therefore E is not the center of the circle ABC. in like manner it may be demonstrated that no other point but D is the center; D therefore is the center. Wherefore if a point be taken, &c. Q. E. D.

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a. 7. 3.

PRO P. X. THEO R.

ONE circumference of a circle cannot cut another

in more than two points.

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circle DEF. but K is also the center of the circle ABC; therefore Book III.
the same point is the center of two circles that cut one another,
which is impossible b. Therefore one circumference of a circle b. s. i.
cannot cut apother in more than two points. Q. E. D.

PROP. XI. THE O R.
F two circles touch each other internally, the straight

line which joins their centers being produced shall
pass through the point of contact.

מן

GE

2. 20. I.

Let the two circles ABC, ADE touch each other internally in
the point A, and let F be the center of the circle ABC, and G the
center of the circle ADE. the straight
line which joins the centers F, G be-

Α.
ing produced passes thro' the point A.
For if not, let it fall otherwise, if

H
possible, as FGDH, and join AF, AG.
and because AG, GF are greater
than FA, that is than FH, for FA is
equal to FH, both being from the

C

E
same center ; take away the common
part FG, therefore the remainder AG

B
is greater than the remainder GH. but AG is equal to GD, there-
fore GD is greater than GH, the less.than the greater, which is
impossible. therefore the straight line which joins the points F, G
cannot fall otherwise than upon the point A, that is, it must pass
thro' it. Therefore if two circles, &c. Q. E. D.

1

PRO P. XII. THE O R.

IF
F two circles touch each other externally, the straight

line which joins their centers shall pass thro' the
point of contact.

Let the two circles ABC, ADE touch each other externally in the point A ; and let F be the center of the circle ABC, and G the center of ADE. the straight line which joins the points F, G shall pass thro' the point of contact A.

For if not, let it pass otherwise, if possible,.as FCDG, and join

Book III. FA, AG. and because F is the center of the circle ABC, AF-is equal to FC. also because

E
G is the center of the cir-

B
cle ADE, AG is equal to
GD. therefore FA, AG

A
are equal to FC, DG ;
wherefore the whole FG

F
is greater than FA, AG.
but it is also less *; which
is impossible. therefore the straight line which joins the points F,
G shall not pass otherwise than thro' the point of contact A, that
is, it must pass thro' it. Therefore if two circles, &c. Q. E. D.

PRO P. XIII. THEO R.

Bee N.

ONE

NE circle cannot touch another in more points than one,

whether it touches it on the inside or outside.

For, if it be possible, let the circle EBF touch the circle ABC

in more points than one, and first on the inside, in the points B, D; 4. 10. 11. 1.join BD, and draw 'GH bisecting BD at right angles. therefore because the points B, D are in the circumference of each of the

H
А.

E
B

A
E

H B

b. 2. 3.

G circles, the straight line BD falls within 6 each of them. and their c. Cor. 1. 3. centers are in the straight line GH which bisects BD at right

angles; therefore GH passes thro' the point of contact d. but it does not pass thro' it, because the points B, D are without the straight line GH, which is absurd. therefore one circle cannot touch another on the inside in more points than one.

Nor can two circles touch one another on the outside in more

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