c

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

C. 23. I.

d. 4. I.

then because GE is equal to EH, and
EF common to the two triangles GEF,
HEF; the two fides GE, EF are equal
to the two HE, EF; and the angle
GEF is equal to the angle HEF, there-
fore the bafe FG is equal to the
bafe FH. but befides FH no other
ftraight line can be drawn from F to
the circumference equal to FG. for

A

B

DH

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

IF

PROP. VIII. THE OR.

F any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one paffes thro' the center; of those which fall upon the concave circumference the greateft is that which paffes thro' the center; and of the reft, that which is nearer to that thro' the center is always greater than the more remote. but of thofe which fall upon the convex circumference, the leaft is that be tween the point without the circle, and the diameter; and of the reft, that which is nearer to the least is always lefs than the more remote. and only two equal ftraight lines can be drawn from the point unto the circumference, one upon each fide 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 paffes thro' the center. of those which fall upon the concave part of the circumference AEFC, the greatest is AD which paffes 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

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

D

24.

Take M the center of the circle ABC, and join ME, MF, MC, 1. 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. than ED, therefore alfo 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, MD; but the angle EMD is greater than the angle FMD, therefore the base ED is greater than the base FD. in like manner it may be shewn that FD is greater than CD. therefore DA is the greateft; and DE greater than DF, and DF than DC. and because MK, KD are greater than MD. and MK is equal to MG, the remainder KD is greater than the remainder GD, that is, GD is lefs than KD. and because MK, DK

are drawn to the point K within the F

H

GB

N

M

d. 4. Azi

triangle MLD from M, D the ex

tremities of its fide MD; MK, KD

E

A

are less than ML, LD, whereof

c. 21. Es

MK is equal to ML, therefore the remainder DK is lefs than the
remainder DL. in like manner it may be shewn that DL is less than
DH. therefore DG is the leaft, and DK lefs than DL, and DL than
DH. Also there can be drawn only two equal ftraight 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 fides
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.
bafe DB. but befides DB there can be no ftraight 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 alfo to DB, therefore DB
is equal to DN, that is the nearer to the leaft equal to the more
remote, which is impoffible. If therefore any point, &c. Q. E. D.

Book III.

2. 7. 3.

2.9.3.

IF

F a point be taken within a circle, from which there fall more than two equal ftraight 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,

DE

G

join DE and produce it to the cir-
cumference in F, G; then FG is a
diameter of the circle ABC. and be-
caufe in FG the diameter of the F
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
DB, and DB than DA. but they are
likewife equal, which is impoffible. therefore E is not the center
of the circle ABC. in like manner it may be demonftrated that no
other point but D is the center; D therefore is the center. Where-
fore if a point be taken, &c. Q. E. D.

A B

circle DEF. but K is alfo the center of the circle ABC; therefore Book III. the fame point is the center of two circles that cut one another,

which is impoffible. Therefore one circumference of a circle b. s. j. cannot cut another in more than two points. Q. E. D.

PROP. XI. THEOR.

IF
two circles touch each other internally, the ftraight
line which joins their centers being produced shall
pass through the point of contact.

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 ftraight
line which joins the centers F, G be-
ing produced paffes thro' the point A.

For if not, let it fall otherwife, if
poffible, 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
fame center; take away the common
part FG, therefore the remainder AG

H

GF

E

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
impoffible. 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.

IF

PROP. XII. THE OR.

2. 20. I.

two circles touch each other externally, the ftraight line which joins their centers fhall 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 poffible, as FCDG, and join

Book III. FA, AG. and because F is the center of the circle ABC, AF is

Bee N.

but it is alfo lefs; which

is impoffible. therefore the straight line which joins the points F, G fhall not pafs otherwise than thro' the point of contact A, that is, it must pass thro' it. Therefore if two circles, &c. Q. E. D.

ON

PROP. XIII. THEOR.

NE circle cannot touch another in more points than whether it touches it on the infide or outfide.

one,

For, if it be poffible, let the circle EBF touch the circle ABC in more points than one, and firft on the infide, in the points B, D;

. 10. 11. 1. join BD, and draw GH bifecting BD at right angles. therefore because the points B, D are in the circumference of each of the

b. 2. 3.

circles, the ftraight line BD falls within beach of them. and their c. Cor. 1. 3. centers are in the straight line GH which bifects BD at right

d. 1. 3.

་་་

angles; therefore GH paffes thro' the point of contact. but it does not pass thro' it, because the points B, D are without the straight line GH, which is abfurd. 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