Sidebilder
PDF
ePub

99

[blocks in formation]

IF

PROPOSITION IV. THEOREM III.

F in a circle (ADCB) two chords (AC, DB,) cut one another, they are divided into two unequal parts.

Hypothefis.

The two chords AC, DB, of the O ADCB

cut one another in the point E.

Thefis.

Thefe chords are divided into two unequal parts.

If not,

DEMONSTRATION.

The chords AC, DB, bife&t one another.

Preparation.

From the center F to the point E, draw the portion of the dia-
meter FE.

BECAUSE the diameter, or its part FE, bifects each of the chords

AC, DB, of the O ADCB (Sup.).

1. This ftraight line FE is upon each of the chords AC, DB.

Pof. I.

P. 3. B. 1.

2. Confequently, the V FEB, FEA, are to one another; which S Ax.10.B.1. is impoffible.

3. Wherefore, the two chords AC, DB, are divided into two unequal

Ax. 8. B. 1.

parts.

Which was to be demonftrated.

B

IF

[ocr errors]

PROPOSITION V. THEOREM IV.

F two circles (ABE, ADE,) cut one another, they shall not have the fame center (C).

[blocks in formation]

1. From the point C to the point of fection A, draw the ray CA. } Pos. 1.

cuts the two O in D & B.

BECAUS

ECAUSE the ftraight lines CA, CD, are drawn from the center C

to the O ADE (Prep. 1. & 2.).

1. These straight lines CA, CD, are = to one another.

It is proved in the fame manner, that:

2. The ftraight lines CA, CB, are to one another.

3. Confequently, CB will be to CD; which is impossible.

4. Therefore, the two circles ABE, ADE, have not the same center.

Which was to be demonstrated,

D. 15. B. I.

Ax. 8. B. 1.

[blocks in formation]

IF

PROPOSITION VI. THEOREM V.

F two circles (BCA, ECD,) touch one another internally in (C); they shall not have the fame center (F).

[blocks in formation]

Draw the rays FB, FC.

BECAUSE

ECAUSE the point F is the center of the BCA (Sup.).

1. The rays FB, FC, are to one another.

Again, the point F being alfo the center of O ECD (Sup.)

2. The rays FE, FC, are to one another.

=

[merged small][ocr errors][merged small][merged small]

Which was to be demonftrated.

3. Confequently, FB FE (Ax. 1. B. 1.); which is impoffible. 4. Wherefore, the two O BCA, ECD, have not the fame center.

[blocks in formation]

I

PROPOSITION VII.

F any point (F) be taken in a circle (AHG) which is not the center (E); of all the ftraight lines (FA, FB, FC, FH,) which can be drawn from it to the circumference, the greatest is (FA) in which the center is, & the part (FD) of that diameter is the least, & of any others, that (FB or FC) which is nearer to the line (FA) which passes thro' the center is always greater than one (FC or FH) more remote, & from the fame point (F) there can be drawn only two ftraight lines (FH, FG,), that are equal to one another, one upon each fide of the shortest line (FD).

[blocks in formation]

1.THE

HE two fides FE + EB of the ▲ FEB are > the third FB.

But EB is to EA (D. 15. B. 1.).

2. Therefore, FE + EA, or FA is > FB.

It is proved in the fame manner that:

3. The ftraight line FA, is the greatest of all the straight lines drawn

from the point F to the O AHG.

Which was to be demonftrated. I.

P. 20. B. 1.

4. Again, the two fides FE + FH of the ▲ FEH are > the third EH. P. zó, B. 1, And ED being to EH (D. 15. B. 1.).

[blocks in formation]

7. The ftraight line FD, which is the produced part of FA, is the leaft of all the ftraight lines drawn from the point F to the O AHG. Which was to be demonstrated. II.

Moreover, the fide FE being common to the two ▲ FEB, FEC, the fide EB the fide EC (D. 15. B. 1.), & the V FEB > V FEC (Ax. 8. B. 1.).

8. The base FB will be > the base FC.

For the fame reafon :

9. The ftraight line FC is > FH.

10. Confequently, the ftraight line FB or FC which is nearer the line FA, which paffes thro' the center, is > FC or FH more remote.

[blocks in formation]

1. Make V FEG = to V FEH, & produce EG until it meets
the AHG.

2. From the point F to the point G, draw the straight line FG.

Then, EF being common to the two ▲ FEH, FEG, the fide EH the fide EG (D. 15. B. 1.), & the \ FEH = to the V FEG (II. Prep. 1.).

11. The base FH will be to the bafe FG.

But because any other ftraight line, different from FG, is either nearer the line FD, or more remote from it, than FG.

12. Such a ftraight line will be alfo < or > FG (Arg. 10.).

13. Wherefore, from the fame point F, there can be drawn only two ftraight lines FH, FG, that are to one another, one upon each fide of the fhortest line FD.

Which was to be demonftrated. IV.

P. 24. B. 1.

P. 23. B. L
Pof. 1.

P. 4. B. 1.

« ForrigeFortsett »