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

the fquare of DB.

Either DCA paffes thro' the center, or it does not; firft, let it

pafs thro' the center E, and join EB;

D

therefore the angle EBD is a right

angle., and because the 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 toge

c

a. 18. 3.

b. 6. 2.

E

A

ther 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 fquare of the tangent DB.

C. 47. I.

But if DCA does not pafs thro' the center of the circle ABC, take the center E. and draw EF perpendicular to AC, and d. 1. 3. join EB, EC, ED; then EFD is a right angle. and because thee. 12. 1. ftraight line EF, which paffes thro' the center, cuts the straight line AC, which does not pafs thro' the center, at right angles, it fhall likewife bifect fit; therefore AF is equal to FC.

and because the ftraight line AC is bi-
fected in F, and produced to D, the rec-
tangle AD, DC together with the fquare
of FC, is equal to the fquare of
FD. to each of thefe equals add the B
fquare of FE, therefore the rectangle
AD, DC together with the fquares of
CF, FE, is equal to the fquares of DF, A
FE. but the fquare of ED is equal to
the fquares of DF, FE, because EFD is

C

a right angle; and the fquare of EC is

F

D

f. 3. 3.

equal to the fquares of CF, FE; therefore the rectangle AD, DC together with the fquare of EC, is equal to the fquare of ED. and CE is equal to EB, therefore the rectangle AD, DC toger,

C. 47. I.

Book III. ther with the fquare of EB, is equal to the fquare of ED. but in the fquares of EB, BD are equal to the fquare of ED, 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 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

E

C

B

PROP. XXXVII. THEOR.

IF 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 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.

c

Draw the ftraight line DE touching the circle ABC, find its 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 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 ftraight line DB. and FE is equal to FB,

wherefore DE, EF are equal to
DB, BF; and the base FD is com-
mon to the two triangles DEF,
DBF; therefore the angle DEF is
equal to the angle DBF, but
DEF is a right angle, therefore alfo
DBF is a right angle, and FB, if
produced, is a diameter, and the B
ftraight line which is drawn at
right angles to a diameter, from
the extremity of it, touches the
circle. therefore DB touches the
circle ABC. Wherefore, if from
a point, &c. Q. E. D.

e

Book IV.

THE

ELEMENTS

OF

EUC L
CLI D.

BOOK IV.

DEFINITION S.

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 in

fcribed, each upon each.

II.

In like manner, a figure is faid to be defcribed

about another figure, when all the fides of the

circumfcribed figure pafs thro' the angular

points of the figure about which it is described, each thro each.

III.,

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

circle.

IV.

A rectilineal figure is faid to be described 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 inscribed
in a rectilineal figure, when the circumfe-
rence of the circle touches each side of the
figure.

VI.

95

Book IV.

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

VII.

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

PROP. I. PROB.

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

the circle.

Let ABC be the given circle, and D the given ftraight 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 ftraight line BC is placed equal to. D. but if it is not, BC is

A

CE; 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 ftraight line D, which is not greater than the diameter of the circle. Which was to be done.

IN

PROP. II. PROB.

Na given circle to infcribe a triangle equiangular to a given triangle.