to the squares of AG, GF: wherefore the rectangle contained by AE, EC together with the Square of FE is equal to the fquare of FA: but FA is equal to FD; wherefore the rectangle contained by AE, EC together with the Square of FE is equal to the square of FD. Certainly for the fame reason also the rectangle contained by DE, EB together with the Square of FE is equal to the fquare of FD but it has also been demonftrated that the rectangle contained by AE, EC together with the fquare of FE is equal to the fquare of FD: wherefore the rectangle contained by AE, EC together with the square of FE is equal to the rectangle contained by DE, EB together with the square of FE: let the square of FE which is common be taken away; therefore the remaining rectangle contained by AE, EC is equal to the remaining rectangle contained by DE, EB.

Wherefore if in a circle two ftraight lines cut one another, the rectangle contained by the fegments of the one is equal to the rectangle contained by the fegments of the other. Which was to be demonstrated.

If any point be taken without a circle, and two straight lines fall from it upon the circle; and one of them cuts the circle and the other touches it; the rectangle contained by the whole cutting line and the fegment without, taken between the point and the convex circumference, will be equal to the square of the touching line. For let D, any point without the circle ABC, be taken; and from the point D, let the two straight lines DCA, DB fall upon

Book III. the circle ABC; and let DCA cut the circle ABC; and let DB touch it: I fay that the rectangle contained by AD, DC is equal to the fquare of DB: the ftraight line DCA either passes through the center or not.

First let it pass through the center; and let F be the center of the circle ABC; and let FB be joined; therefore (by 18. 3.) FBD is a right angle: And because the ftraight line AC hath been cut in halves in the point F, and CD is added to it; therefore (by 6. 2.) the rectangle contained by AD, DC together with the square of FC is equal to the Square of FD: but the Square of FD (by 47· 1.) is equal to the fquares of FB, BD; for the angle FBD is a right angle: wherefore the rectangle contained by AD, DC together with the Square of FB is equal to the Squares of FB, BD; let the common Square of FB be taken away; therefore the remaining rectangle contained by AD, DC is equal to the remaining Square, of DB the touching line.

B

A

C

B

F

F

E

A

But let DA not pafs through the center of the circle ABC: and let E the center of the circle be taken; and from E let EF be drawn perpendicular to AC; and let EB, EC, ED be joined; wherefore. EFD is a right. angle; and because EF a certain ftraight line through the center, cuts at right angles AC a certain ftraight line not through the center, it will alfo (by 3. 3.) cut it in halves; therefore AF is equal to FC: and because the straight line AC hath been cut in halves at F; and CD is added to it; therefore (by 6. 2.) the rectangle contained by AD, DC together with the Square of FC is equal to the square of FD: let the common Square of FE be added; therefore the rectangle contained by AD, DC together with the fquares of CF, FE are equal to the fquares of DF, FE: but the fquare of DE is equal: to the fquares of DF, FE (by 47. 1.); for EFD is a right angle; and the square of CE is equal to the fquares of CF, FE: Wherefore the rectangle contained by AD, DC together with the fquare of EC is equal to the fquare of ED: but CE is equal to EB; therefore, the rectangle contained by AD, DC together with the

fquare

square of EB is equal to the fquare of ED: but the fquare of ED Book IM. is equal to the squares of EB, BD; for the angle EBD is a right angle wherefore the rectangle contained by AD, DC together with the square of EB is equal to the fquares of EB, BD; let the common fquare of EB be taken away; therefore the remaining rectangle contained by AD, DC is equal to the remaining fquare of

DB.

Wherefore if any point be taken without a circle, &c. Which was to be demonftrated.

If any point be taken without a circle, and two ftraight lines fall from the point upon the circle, and one of them cuts the circle and the other meets it, and if the rectangle contained by the whole cutting line, and the fegment without, taken between the point and the convex circumference, be equal to the Square of the line which meets it; the meeting line will touch the circle.

For let D be taken any point without the circle ABC; and let the two ftraight lines DCA, DB fall from the point D, upon the circle ABC; and let DCA cut the circle, and DB meet it; and let the rectangle contained by AD, DC be equal to the Square of DB; I say that DB touches the circle ABC.

For let DE be drawn (by 17.3.) touching the circle ABC; and let F the center of the circle ABC be taken (by 1.3.); and let FE, FB, FD be joined therefore the angle FED is (by 18. 3.) a right angle.

B

And fince DE touches the circle ABC; and DCA cuts it; therefore (by 36. 3.) the rectangle contained by AD, DC is equal to the Square of DE; but the rectangle contained by AD, DC is fuppofed equal to the Square of DB; therefore the fquare of DE is equal to the fquare of DB; wherefore DE is equal to DB; and FE is alfo A equal to FB; therefore the two DE, EF are equal

D

E

F

to the two DB, BF, and DF is a common base to them; therefore (by 8. 1.) the angle DEF is equal to the angle DBF; but DEF is

a right

Book III. a right angle; therefore DBF is also a right angle; and FB produced is a diameter; but the ftraight line drawn from the extremity at right angles to the diameter touches the circle ABC (by cor. to 16. 3.); certainly it will be demonftrated in the same manner if the center happen to be in AC.

Wherefore if any point be taken without a circle, &c. Which was to be demonstrated.

I.

DEFINITION S.

A Rectilineal figure is faid to be infcribed in a rectilineal fi- Book IV. gure, when each of the angles of the infcribed figure touches each fide of the figure in which it is infcribed. 2. And in like manner a figure is faid to be circumfcribed about a figure, when each fide of the circumfcribed figure touches each angle of the figure about which it is circumfcribed. 3. And a rectilineal figure is faid to be inscribed in a circle when each angle of the inscribed figure touches the circumference of the circle. 4. And a rectilineal figure is faid to be circumfcribed about a circle, when each fide of the circumfcribed figure (is a tangent to the circle) touches: the circumference of the circle. 5. In like manner a circle is faid to be infcribed in a figure, when the circumference of the circle touches each fide of the figure within which it is inscribed. 6. But a circle is faid to be circumfcribed about a figure, when the