C 23. I.

d 4. I.

Book III. fhorteft line FD: At the point E in the ftraight line EF, make the angle FEH equal to the angle GEF, and join FH: Ther becaufe GE is equal to EH, and EF common to the two tri angles GEF, HEF; the two fides GE, EF are equal to the twe HE, EF; and the angle GEF is equal to the angle HEF; there fore the base FG is equal to the base FH: But, befides FH, no other ftraight line can be drawn from F to the circumference equal to FG: For, if there can, let it be FK; and becaufe FK is equal to FG, and FG to FH, FK is equal to FH; that is, a line nearer to that which paffes through the centre, 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, whereof one paffes through the centre; of those which fall upon the concave circumference, the greatest is that which paffes through the centre; and of the rest, that which is nearer to that through the centre is always greater than the more remote: But of thofe which fall upon the convex circumference, the leaft 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 straight 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 let the ftraight lines DA, DE, DF, DC be drawn to the cir cumference, whereof DA passes through the centre. Of thos which fall upon the concave part of the circumference AEFC, the greateft is AD which paffes through the centre; 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 leaft is DG between the

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

DH.

C

H

D

GB

N

C 24. I.

M

d 4. Ax.

F

E

A

Take M the centre of the circle ABC, and join ME, MF, a 1. 3. MC, 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 ED; therefore alfo AD is greater than ED: b 20. I. Again, becaufe 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 bafe FD: In 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 becaufe 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 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 than the remainder DL: In like manner it may be fhewn, that DL is less than DH: Therefore DG is the least, and DK less than DL, and DL than DH: Alfo there can be drawn only two equal ftraight lines from the point D to the circumference, one upon each fide of the leaft: At the point M, in the ftraight line MD, make the angle DMB equal to the angle DMK, and join DB: And becaufe 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, befides DB, f 4. I. 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.

PROP.

F 21. I.

Book HI.

a 7. 3.

I'

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 centre of the circle.

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

DE

For, if not, let E be the centre, join DE and produce it to the cir cumference in F, G; then FG is a diameter of the circle ABC: And because in FG, the diameter of the circle ABC, there is taken the point D which is not the centre, DG fhall 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 centre of the circle ABC: In like manner, it may be demonftrated, that no other point but D is the centre; D therefore is the centre. Wherefore, if a point be taken, &c. Q. E. D.

G

A B

a 9.3.

ONE

NE circumference of a circle cannot cut another in more than two points.

77

the centre of the circle DEF: But K is alfo the centre of the Book III. circle ABC; therefore the fame point is the centre of two circles that cut one another, which is impoffible. Therefore one b 5.3. circumference of a circle cannot cut another in more than two points. Q.E. D.

two circles touch each other internally, the ftraight line which joins their centres being produced fhall pafs through the point of contact.

A

Let the two circles ABC, ADE touch each other internally in the point A, and let F be the centre of the circle ABC, and G the centre of the circle ADE: The ftraight line which joins the centres F, G, being produced, paffes through 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 centre; take away the common part FG; therefore the remain

H

G

F

Ca 20,

E

B

der AG 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 otherwife than upon the point A, that is, it must país through it. Therefore, if two circles, &c. QE. D.

IF

PROP. XII. THEOR.

two circles touch each other externally, the straight line which joins their centres fhall pass through the point of contact.

Let the two circles ABC, ADE touch each other externally in the point A; and let F be the centre of the circle ABC, and G the centre of ADE: The ftraight lines which joins the points F, G fhall pass through the point of contact A.

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

Book III. join FA, AG: And because F is the centre of the circle ABC,

a 20. I.

AF is equal to FC: Also,
because G is the centre of
the circle ADE, AG is c-
qual to GD: Therefore
FA, AG are equal to FC,
DG; wherefore the whole
FG is greater than FA,
AG: But it is also lefs a;
which is impoffible :
Therefore the ftraight

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

See N.

Ο

NE circle cannot touch another in more points than 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 firft on the infide, in the

a 10. it. 1. points B, D; join BD, and draw a GH bifecting BD at right angles: Therefore, becaufe the points B, D are in the circumfe

b 2. 3.

rence of each of the circles, the ftraight line BD falls within beach of them: And their centres are in the straight line GH c Cor. 1.3. which bifects BD at right angles; therefore GH paffes through the point of contact ; but it does not pafs through 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.

d II. 3.

Nor can two circles touch one another on the outfide in

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