Book III. FD. at the point E in the straight line EF, make the angle FEH

equal to the angle GEF, and join

c. 23. 1. FH. then because GE is equal to

EH, and EF common to the two tri

c

BA

rence 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 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 any point be taken without a circle, and straight lines

be drawn from it to the circumference, paffes thro' the center; of thofe which fall

whereof one upon the con

cave circumference the greatest 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 those which fall upon the convex circumference, the least is that between the point without the circle, and the diameter; and of the reft, that which is nearer to the leaft 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 least.

Let ABC be a circle, and D any point without it, from which fet the straight lines DA, DE, DF, DC be drawn to the circumfetence, whereof DA passes 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 least

is

is DG between the point D and the diameter AG; and the nearer Book III. to it is always lefs 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, a. 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 than b. so. ti 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, MD; but the

is

F

D

KG B

H

N

E

A

ċ. 24. śl

d. 4. At.

M

c. at. 1.

angle EMD is greater than the angle FMD, therefore the bafe ED than the base FD. in greater like manner it may be fhewn 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 C remainder GD, that is, GD is lefs than KD. and because MK, DK are drawn to the point K within the triangle MLD from M, D the extremities of its fide MD; MK, KD are lefs than ML, LD, whereof MK is equal to ML, therefore the remainder DK is lefs tlian the remainder DL. in like manner it may be fhewn 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 straight lines from the point D to the circumference, one upon each fide of the leaft. 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 to the bafe DB. but f. 4. 11 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 becaufe 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.

IF

PROP. IX. THEOR.

[F a point be taken within a circle, from which there fall more than two equal straight lines to the circum ference, 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 becaufe in FG the diameter of the circle ABC there is taken the point D which is not the center, DG shall be the greatest line from it to the .7.3. circumference, and DC greater than DB, and DB than DA. but they

a

F

DE

G

A B

are likewife equal, which is impoffible. therefore E is not the cen ter of the circle ABC. in fike 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.

circle DEF. but K is also 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 can- b. 5. 3. not cut another in more than two points. Q.E. D.

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 straight line which joins the centers F, G being produced paffes thro' the point A.

For if not, let it fall otherwise, 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

A

H

GE

a. 20. 1.

E

B

is greater than the remainder GH. but AG is equal to GD, therefore 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

F two circles touch each other externally, the straight 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 ftraight line which joins the points F, G fhall pafs thro' the point of contact A.

For if not, let it pafs otherwife, if poffible, as FCDG, and join

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

See

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

PROP. XIII. THEOR.

N. ONE circle cannot touch another in more points than Ο

one, whether it touches it on the infide or outfide,

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

9. 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.

C G

circles, the straight line BD falls within each of them. and their c. Cor. 1.3. centers are in the straight line GH which bisects BD at right and. 11.3. gles; 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 infide in more points than one.

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

than