centre G* Therefore the angle ADG is equal to the angle Book III. GDB: But when a straight line standing upon another straight C 8. 1.

F

C

line makes the adjacent angles equal to one
another, each of the angles is a right
angled: Therefore the angle GDB is a
right angle: But FDB is likewise a right
angle; wherefore the angle FDB is equal
to the angle GDB, the greater to the
less, which is impoffible: Therefore G A
is not the centre of the circle ABC:
In the fame manner it can be shewn, that
no other point but F is the centre; that
is, F is the centre of the circle ABC: Which was to be found.
COR. From this it is manifeft, that if in a circle a straight
line bifect another at right angles, the centre of the circle is in
the line which bifects the other.

PROP. II. THEOR.

E

D

dio.def.1.

F any two points be taken in the circumference of See N. a circle, the ftraight line which joins them fhall

IF

fall within the circle.

Let ABC be a circle, and A, B any two points in the cir cumference; the ftraight line drawn from A to B fhall fall within the circle.

b

D

a 1. 3.

E

B

b. 5. 1.

Take any point E in AB, and find a D the centre of the circle, and join AD, DB, DE, and let DE meet the circumference in F: Then, because DA is equal to DB, the angle DAB is A equal to the angle DBA; and because AE, a fide of the triangle DAE, is produced to B, the angle DEB is greater than the angle DAE; c 16. 1. but DAE is equal to the angle DBE; therefore the angle DEB is greater than the angle DBE: But to the greater angle the greater fide is oppofited; DB is therefore greater than DE: But DB is equal to DF; wherefore DF is greater than DE; the point E is therefore within the circle: and E is any point in the ftraight line AB; therefore AB falls within the circle ABC. Wherefore, if any two points, &c. Q.E. D.

d 19. I.

PROP.

*N. B. Whenever the expreffion "ftraight lines from the centre," or "drawn from the centre," occurs, it is to be understood, that they are drawn to the circumference.

BOOK III.

a 1. 3.

b 8. I.

IF

PROP. III. THEOR.

F a flraight line drawn through the centre of a circle bifect a ftraight line in it which does not pass through the centre, it fhall cut it at right angles; and, if it cuts it at right angles, it fhall bifect it.

Let ABC be a circle; and let CD, a straight line drawn through the centre, bifect any straight line AB, which does not pass through the centre, in the point F: It cuts it also at right angles.

b

Take 2 E the centre of the circle, and join EA, EB: Then, because AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two fides in the one equal to two fides in the other, and the base EA is equal to EB; therefore the angle AFE is equal to BFE But when a ftraight line ftanding upon another makes the adjacent angles c1o. def. 1. equal, each of them is a right angle: Therefore each of the angles AFE, BFE is a right angle; wherefore the ftraight line CD, drawn through the centre bifecting another AB that does not pafs through the centre, cuts the fame at right angles.

c

2

F

D

But let CD cut AB at right angles; CD alfo bifsects it, that is, AF is equal to FB.

d

The fame conftruction being made, because EA, EB from d 5. 1. the centre are equal to one another, the angle EAF is equal to the angle EBF; and the right angle AFE is equal to the right angle BFE: Therefore, in the two triangles EAF, EBF, there are two angles in one equal to two angles in the other, and the fide EF, which is oppofite to one of the equal angles € 26. 1. in each, is common to both; therefore the other fides are equal; AF therefore is equal to FB. Wherefore, if a straight line, &c. Q. E. D.

PROP. IV. THEOR.

F in a circle two ftraight lines cut one another which do not both pafs through the centre, they

do not bifect each the other.

Let ABCD be a circle, and AC, BD two ftraight lines in it which cut one another in the point E, and do not both pass through the centre: AC, BD do not bifect one another.

B

a 1. 3.

F

D

E.

C

b 3.3.

If one of the lines pafs through the centre, it is plain that it Book III. cannot be bifected by the other which does not pass through the centre: But, if neither of them pass through the centre, let, if poffible, AE be equal to EC, and BE to ED, and take F the centre of the circle, and join EF: And because FE, a ftraight line through the centre, bifects another AC which does not pafs through the centre, it shall cut it at right angles; wherefore FEA is a right angle: Again, because the straight line FE bifects the straight line BD which does not pass through the centre, it fhall cut it at right angles; wherefore FEB is a right angle: And FEA was shewn to be a right angle; therefore FEA is equal to the angle FEB, the lefs to the greater; which is impoffible: Therefore AC, BD do not bifect one another. Wherefore, if in a circle, &c. Q. E. D.

PROP. V. & VI. THEOR.

b

F two circles meet one another, they fhall not have See N. the fame centre.

IF

Let the circumferences of the two circles ABC, CDG meet one another in the point C; the circles have not the fame

centre.

For, if it be poffible, let E be their centre: Join EC, and

CE is equal to EG: But CE was fhewn to be equal to EF; therefore
EF is equal to EG, the less to the greater; which is impoffible:
Therefore E is not the centre of the circles ABC, CDG.
Wherefore, if two circles, &c. Q. E. D.

BOOK III.

a 20. I.

IF

PROP. VII. THEOR.

F any point be taken in the diameter of a circle, which is not the centre, of all the flraight lines which can be drawn from it to the circumference, the greateft is that in which the centre is, and the other part of that diameter is the leaft; 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 shortest 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 ftraight 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.

a

Join BE, CE, GE; and because two fides of a triangle are greater than the third, BE, EF are greater than BF; 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; b 24. 1. therefore the bafe BF is greater than the base FC: For the fame reafon, CF is greater than GF: Again, because GF, FE are greater than EG, and EG is equal to ED;

b

a

K

H

D

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 leaft of all the ftraight lines from F to the circumference; and BF is greater than CF, and CF than GF.

Alfo there can be drawn only two equal ftraight lines from the point F to the circumference, one upon each fide of the c 23. 1. fhorteft line FD: At E in EF, make the angle FEH equal to 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

d

to HEF; therefore the base FG is equal to FH: But, befides Book III. FH, no other straight line can be drawn from F to the circumference equal to FG: For, if there can, let it be FK; and be- d 4. 1. caufe 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 more remote; which is impoffible. Therefore, if any point be taken, &c. Q. E. D.

IF

F any point be taken without a circle, and ftraight lines be drawn from it to the circumference, whereof one paffes through the centre; of those which fall upon the concave circumference, the greateft 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 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 leaft.

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 circumference, whereof DA paffes through the centre. Of those which fall upon the concave part of the circumference AEFC, the greatest 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 lefs than the more remote, viz. DK than DL, and DL than DH.

a 1. 3°

Take a M the centre of the circle ABC, and join ME, MF, 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 b 20. 1. 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 FMD; therefore the bafe ED is greater c than FD: In like manner, it may be fhewn, c 24. 3. that FD is greater than CD: Therefore DA is the greatest;

I 2

and