ftraight line AC is bifected in E, and produced to the point D, the rectangle AD, DC, together with the fquare of EC, is equal b to the fquare of ED; and CE is equal to EB; therefore the rectangle AD, DC, together with the fquare of EB, is equal to the fquare of ED: but the fquare of ED is equal to the fquares of EB, BD, because EBD is a right angle: therefore the rectangle AD, DC, together with the fquare of EB, is equal to the fquares of EB, BD: take away the common fquare

of EB; therefore the remaining rectangle AD, DC is equal to the square of the tangent DB.

But if DCA does not pafs through the centre of the circle

Book III

b 6. 2.

C 47. I.

D

£ 3. 3.

ABC, take d the centre E, and draw EF perpendicular e to d 1. 3AC, and join EB, EC, ED: and because the straight line e 12. I. EF, which paffes through the centre, cuts the straight line AC, which does not pass through the centre, at right angles, it fhall likewife bifect fit; therefore AF is equal to FC: and because the straight line AC is bifected in F, and produ ced to D, the rectangle AD, DC, together with the fquare of FC, is equal b to the fquare of FD: to each of thefe equals add the fquare of FE; therefore the rectangle AD, DC, together with the fquares of CF, FE, is equal to the fquares of DF, FE: but the fquare of ED is equal to the fquares of DF, FE be

B

F

caufe EFD is a right angle; and the fquare of EC is equal to the fquares of CF, FE; therefore the rectangle AD, ÔC, together with the fquare of EC, is equal to the square of ED: and CE is equal to EB; therefore the rectangle AD, DC, together with the fquare of EB, is equal to the fquare of ED: but the fquares of EB, BD are equal to the fquare e of ED, because EBD is a right angle; therefore the rectangle AD, DC, together with the fquare of EB, is equal to the

2

fquares

Book III. fquares of EB, BD : take away the common fquare of EB; therefore the remaining rectangle AD, DC is equal to Wherefore, if from any point, &c.

the fquare of DB.

Q. E. D.

COR. If from any point without a circle, there be drawn two ftraight lines cutting it, as AB, AC, the rectangles contained by the whole lines and the parts of them without the circle, are equal to one another, viz. the rectangle BA, AE to the rectangle CA, AF: for each of them is equal to the fquare of the ftraight line AD which touches the circle.

D

E F

B

a 17.3.

b 18. 3. c 36. 3.

F from a point without a circle there be drawn

two ftraight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle be equal to the fquare of the line which meets it, the line which meets fhall touch the circle.

Let any point D be taken without the circle ABC, and from it let two straight lines DCA and DB be drawn, of which DCA cuts the circle, and DB meets it; if the rectangle AD, DC, be equal to the square of DB; DB touches the circle.

C

Draw a the ftraight line DE touching the circle ABC, find the centre F, and join FE, FB, FD; then FED is a right b angle: and because DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal to the fquare of DE; but the rectangle AD, DC is, by hypothefis, equal to the fquare of DB: therefore the fquare of DE is equal to the fquare of DB; and the straight line DE equal to the

straight

D

ftraight line DB: and FE is equal to FB, wherefore DE, Book III. EF are equal to DB, BF; and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF is equal d to the angle DBF; and DEF is a right angle, therefore also DBF is a right angle but FB, if produced, is a diameter, and the ftraight line

:

which is drawn at right angles to a diameter, from the extremity of it, touches e the circle: therefore DB touches the circle ABC. Wherefore, if from a point, &c. Q. E. D.

B

F

E

d 8. x.

ELEMENTS

OF

GEOMETRY.

BOOK IV.

A

DEFINITION S.

I..

Rectilineal figure is faid to be infcribed in another Book IV. rectilineal figure, when all the angles of the infcribed

figure are upon the fides of the figure

in which it is infcribed, each upon

each.

II.

In like manner, a figure is faid to be described about another figure, when all the fides of the circumfcribed figure

pass through the angular points of the figure about which it is described, each through each.

III.

A rectilineal figure is faid to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle.