For, if they can, let it be F; join FC, and draw any Book III. traight line FEB meeting them in E and B; and becaufe F is the centre of the circle ABC, CF is equal to FB; alfo, because F is the centre of the circle CDE, CF is equal to FE: and CF was fhown equal to FB; therefore FE is equal to FB, the lefs to the greater, which is is impoffible; wherefore F is not the centre of the circles ABC, CDE. Q. E. D.

F

E

B

D

Therefore, if two circles, &c.

IF

PROP. VII. THEOR.

F any point be taken in the diameter of a circle. which is not the centre, of all the ftraight lines which can be drawn from it to the circumference, the greatest is that in which the centre is, and the other part of that diameter is the least ; and, of any others, that which is nearer to the line which paffes through the centre is always greater than one more remote: And from the fame point there can be drawn only two ftraight lines that are equal to one another, one upon each fide of the fhorteft line.

Let ABCD be a circle, and AD its diameter, in which let any point F be taken which is not the centre: let the centre 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 least: 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 a than the third, BE, EF are greater than BF; a 20. 1.

but

B

Book III. but AE is equal to EB; therefore AE, EF, that is, AF, is greater than BF: again, because 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; thereb 24. 1. fore the bafe BF is greater b than the bafe FC for the fame reason, CF is greater than GF: again, becaufe 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 greateft, and FD the least of all the straight lines from F to the circumference; and BF greater than CF, and CF than GF.

is

G

K

H

Alfo there can be drawn only two equal ftraight lines from the point F to the circumference, one upon each fide of the shortest line FD: at the point E in the straight line EF, c 23. 1. make the angle FEH equal to the angle GEF, and join FH: 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; therefore the bafe FG is equal to the bafe FH: but befides FH, no straight line can be drawn from F to the circumference 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 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.

d 4.1.

PROP.

Book III.

PROP. VII. THEOR.

IF any point be taken without a circle, and straigh

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 reft, that which is nearer to that through the centre 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 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 through the centre. Of those 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 least is DG between the point D and the diameter AG; and the nearer to it is always lefs than the more remote, viz. DK than DL, and DL than DH.

Take a 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 b than ED; therefore alfo AD is greater b 25. 1. thanED. Again, because ME is equal to MF, and MD

common

€ 24. I.

may

d

KGB

E

A

M

N

Book III. 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 bafe ED is greater than the base FD; In like manner it 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 b than MD, and MK is equal to MG, d 4. Ax. the remainder KD is greater than the remainder GD, that is GD is less than KD: And becaufe MK, DK are drawn to the point K within the triangle MLD from M, D, the extremities of its fide MD; MK, KD are lefse than ML, LD, whereof MK is equal to ML; there fore the remainder DK is less than the remainder DL: In like manner, it may be fhewn that DL is lefs 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 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 because in the triangles KMD, BMD, the fide KM is equal to the fide BM, and MD common to both, and also the angle KMD equal to the angle BMD, the base DK is equal f to the bafe DB. But, befides DB, no ftraight line can be drawn from D to the circumference, equal to DK: for, if there can, let it be DN; then, because DN is equal to DK, and DK equal to DB, DB is equal to DN; that is, the nearer to DG, the leaft, equal to the more remote, which has been fhewn to be impoffible. If, therefore, any point, &c. Q E. D.

€ 21. I.

f 4. 1.

PROP.

1

Book III.

IF

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

For, if not, let E be the centre, join DE and produce it to the circumference 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 a 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.

a 7. 3.

Ο

PRO P. X. THE OR.

NE circumference of a circle cannot cut ano-
ther in more than two points.

If