a 3.3.

Book III. dicular to AC; therefore AG is equal to GC; wherefore ~ the rectangle AE, EC, together with the square of EG, is equal to the square of AG: To each of these equals add the square of GF; therefore the rectangle AE, EC, together with the squares of EG, GF is equal to the squares of AG, GF: But the D

b s. 2.

C 47. I.

squares of EG, GF are equal to the
square of EF; and the squares of

F

AG, GF are equal to the square of
AF: Therefore the rectangle AE, A
EC, together with the square of EF,
is equal to the square of AF; that is,

to the square of FB: But the square

of FB is equal to the rectangle BE,

ED together with the square of EF; therefore the rectangle AE, EC, together with the square of EF, is equal to the rectangle BE, ED together with the square of EF: Take away the common square of EF, and the remaining rectangle AE, EC is therefore equal to the remaining rectangle BE, ED.

Lastly, Let neither of the straight lines AC, BD pass through

the center: Take the center F, and

through E the interfection of the

straight lines AC. DB draw the
diameter GEFH: And because the

rectangle BE, ED is equal to the

G

same rectangle GE, EH; therefore

the rectangle AE, EC is equal to

B

the rectangle BE, ED. Wherefore, if two straight lines, &c.

Q. E. D.

PROP. XXXVI. THEOR.

F from any point without a circle two straight lines be

touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.

Let D be any point without the circle ABC, and DCA, DB two straight lines drawn from it, of which DCA cuts the circle,

circle, and DB touches the fame: The rectangle AD, DC is Book III.

equal to the square of DB.

Either DCA passes through the center, or it does not; first, let it pass through the center E, and join EB; therefore the angle

EBD is a right angle: And because

a 18. 3.

the straight line AC is bisected in E,

:

and produced to the point D, the rect-
angle AD, DC, together with the
square of EC, is equal to the square
of ED, and CE is equal to EB: There-
fore the rectangle AD, DC, together B
with the fquare of EB, is equal to the
square of ED: But the square of ED,
is equal to the squares of EB, BD, be-
caute EBD is a right angle: Therefore
the rectangle AD, DC, together with
the square of EB, is equal to the
squares of EB, BD: Take away the
common square of EB; therefore the
remaining rectangle AD, DC is equal to the square of the

4

D

b 6.2.

C

E

C 47. Г.

A

tangent DB.

But if DCA does not pass through the center of the circle ABC, taked the center E, and draw EF perpendicular to d 1.3. AC, and join EB, EC, ED: And because the straight line EF, e 12. 1. which passes through the center, cuts the straight line AC, which

does not pass through the center, at right

angles, it shall likewise bisect fit; there

fore AF is equal to FC: And because
the straight line AC is bisected in F,

C

F

D

E

and produced to D, the rectangle AD,
DC, together with the square of FC, is
equal to the square of FD: To each of
these equals add the square of FE; there- B
fore the rectangle AD, DC, together
with the squares of CF, FE, is equal to
the squares of DF, FE: But the square
of ED is equal to the squares of DF, A
FE, because EFD is a right angle; and
the square of EC is equal to the
squares of CF, FE; therefore the
rectangle AD, DC, together with the square of EC, is e-
qual to the square of ED: And CE is equal to EB; therefore
the rectangle AD, DE, together with the square of EB, is equal

f 3. 3.

€ 47. I.

Book III. to the square of ED: But the squares of EB, BD are equal to the square of ED, because EBD is a right angle; therefore the rectangle AD, DC, together with the square of EB, is equal to the squares of EB, BD: Take away the common square of EB; therefore the remaining rectangle AD, DC is equal to the square 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 square of the straight line AD which touches the circle.

D

A

EF

a 17. 3.

b 18. 3.

€ 36. 3.

B

C

PROP. XXXVII. THEOR.

I
F from a point without a circle there be drawn two
straight 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 shall 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 square of DB; DB touches the circle.

Draw the straight 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 square 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 square of DB, and the straight line DE equal to the straight line DB:

And

and FE is equal to FB, wherefore DE, EF are equal to DB, Book III.

BF; and the base FD is common to

the two triangles DEF, DBF; there

D

d 8. 1.

fore the angle DEF is equal d 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 straight line which is drawn
at right angles to a diameter, from the B
extremity of it touches the circle:
Therefore DB touches the circle ABC.

Wherefore, if from a point, &c.
Q.E. D.

C

E

с 16.

F

A

102 Book IV.

THE

ELEMENTS

OF

EUCLID.

A

воок IV.

DEFINITION S.

I.

Rectilineal figure is faid to be inscribed in another recti
lineal figure, when all the angles of the inscribed figure

are upon the fides of the figure in which it is
infcribed, each upon each.

II.

In like manner, a figure is said to be described
about another figure, when all the fides of
the circumscribed figure pass through the an-
gular points of the figure about which it is
described, each through each.

III.

A rectilineal figure is said 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 said to be described about a circle, when

each side of the circumscribed figure
touches the circumference of the circle.

V.

In like manner, a circle is said to be infcri-
bed in a rectilineal figure, when the cir-
cumference of the circle touches each
fide of the figure.

VI.