DB touches the fame. the rectangle AD, DC is equal to the fquare Book III. of DB.

a. 18.3.

b. 6. 2.

C

B

Either DCA paffes thro' the center, or it does not; first, let it país thro' the center E, and join EB; therefore the angle EBD is a right angle. and because the straight line AC is bifested in E, and produced to the point D, the rectangle AD, DC together with the fquare of EC is equal b 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 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

A

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

e

C. 47. L

But if DCA does not pafs thro' the center of the circle ABC, taked d. 1. 3. the center E, and draw EF perpendicular to AC, and join EB, e. 13. I. EC, ED; then EFD is a right angle. and because the straight line

B

D

f. 3. 3.

F

E

EF which paffes thro' the center, cuts
the straight line AC, which does not
pass thro' the center, at right angles, it
shall likewise bisect fit; therefore AF is
equal to FC. and because the straight
line AC is bifected in F, and produced to
D, the rectangle AD, DC together with
the fquare of FC is equal b to the square
of FD. to each of these equals add the
fquare of FE, therefore the rectangle
AD, DC together with the squares of
CF, FE is equal to the fquares of DF,
FE. but the fquare of ED is equal to
the squares of DF, FE, because EFD is a right angle; and the
fquare of EC is equal to the fquares of CF, FE; therefore the rect-
angle AD, DC together with the fquare of EC is equal to the square
of ED. and CE is equal to EB, therefore the rectangle AD, DC to-
gether with the fquare of EB, is equal to the fquare of ED. but the

fquares

c

Book III. fquares of EB, BD are equal to the fquare of ED, because EBD is a right angle; therefore the rectangle AD, DC together with c. 47. 1. 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 fquare of DB. Wherefore if from any point, &c. Q. E. D.

COR. If from any point without a circle there be drawn two straight 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 straight line AD which touches the circle,

D

A

a. 17.3.

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 square 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 ftraight 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 fquare of DB; DB touches the circle.

Draw the straight line DE touching the circle ABC, find its b. 18. 3. center F, and join FE, FB, FD; then FED is a right angle. and because DE touches the circle ABC, and DCA cuts it, the rectangle c. 36. 3. AD, DC is equal to the fquare of DE. but the rectangle AD, DC is, by Hypothesis, equal to the square of DB; therefore the square of DE is equal to the fquare of DB, and the ftraight line DF

c

equal

equal to the straight line DB. and FE is equal to FB, wherefore Book III. DE, EF are equal to DB, BF; and the base FD is common to the two triangles DEF, DBF; therefore the

D

F

E

d. 8. 1.

e. 16. 3

THE

Book IV.

THE

ELEMENTS

OF

EUCLID.

A

BOOK IV.

DÉFINITIONS.

I.

Rectilineal figure is faid to be infcribed in another rectilineal figure, when all the angles of the infcribed figure are upon the fides of the figure in which it is inscribed,

each upon each.

II.

In like manner a figure is faid to be defcribed a-
bout another figure, when all the fides of the
circumfcribed figure pafs thro' the angular

points of the figure about which it is defcribed, each thro' each:

III.

A rectilineal figure is faid to be inscribed in

a circle, when all the angles of the in-
scribed figure are upon the circumference

of the circle.

IV.

A rectilineal figure is faid to be defcribed about a circle, when each fide of the circumfcribed figure touches the

circumference of the circle.

V.

In like manner a circle is faid to be infcribed

in a rectilineal figure, when the circum-
ference of the circle touches each fide of
the figure.

VI. Á

95 Book IV.

VI.

A circle is faid to be defcribed about a rectilincal figure, when the circumference of the circle paffes thro' all the angular points of the figure about which it is described.

VII.

Aftraight line is faid to be placed in a circle, when the extremities of it are in the circumference of the circle.

PROP. I. PROB.

a

a given circle to place a ftraight line, equal to a given straight line not greater than the diameter of the circle.

Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle.

Draw BC the diameter of the circle ABC; then, if BC is equal to D, the thing required is done; for in the circle ABC a straight line BC is placed equal to D. but

a

if it is not, BC is greater than D; make CE equal to D, and from the center C, at the diftance CE defcribe the circle AEF, and join CA. therefore because C is the center of the circle AEF, CA is equal to CE;

D

A

but D is equal to CE, therefore D is equal to CA. wherefore in the circle ABC a straight line is placed equal to the given straight line D, which is not greater than the diameter of the circle. Which was to be done.

PROP. II. PROB.

IN a given circle to infcribe a triangle equiangular to a given triangle,

Let