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Book III. the circle, and DB touches it; the

rectangle AD.DC is equal to the

D

square of DB.

Let it pass

2 18. 3.

E

b 6. 2.
c 47. 1.

d 1. 3. e 12. 1.

Either DCA passes through the
centre, or it does not.
through the centre E, and join
EB; therefore the angle EBD is a B
righta angle. Because the straight
line AC is bisected in E, and pro-
duced to the point D, AD.DC+EC
=bED. But EC=EB, therefore
AD.DC+EB=ED. Now ED=C
EB2+BD, therefore AD.DC+EB2
=EB2+BD2, therefore if EB' be taken

A
from each, AD.DC=BD?.

But if DCA do not pass through the centre of the circle
ABC, taked the centre E, and draw

D
EF perpendiculare to AC, and join
EB, EC, ED. Because the straight
line EF, which passes through the
centre, cuts the straight line AC,
which does not pass through the
centre, at right angles, it likewise В.
bisects it; therefore AF is equal to
FC. Because the straight line AC

F
is bisected in F, and produced to D,
DAD.DC+FC2=FD. Add FE2 to
both, then AD.DC+FC+FE2=
FD2+FE2. But ECP=FC2+FE,
and ED=FD2+ FE ; therefore
AD.DC+EC=ED?. Now EDP=EBP+BD=EC2+BD2;
therefore AD.DC+EC2=EC2+BD2; therefore AD.DC=
BD2. Wherefore, if from any point, &c. Q. E. D.

f 3. 3.

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IF from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; and 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 will touch the circle,

Book III.

a 17. 3.

b 18. 3.

c 36. 3.

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.

Drawa the straight line DE touching the circle ABC; find the centre F, and join FE, FB, FD; then FÉD is a rightb 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, B equal to the square of DB; therefore the square of DE is equal to the square of DB; therefore the

F straight line DE is equal to the straight line DB. Because FE is equal to FB, and DE equal to DB, and the base FD common to the two triangles DEF, DBF, the angle DEF is equals to the angle DBF. But DEF is a right angle, therefore also DBF is a right angle. Now FB, if produced, will be a diameter, therefore DB touchese the circle ABC. Wherefore, if from a point, &c. Q. E. D.

d 8. 1.

e 16. 3.

ELEMENTS OF GEOMETRY.

BOOK IV.

DEFINITIONS,

I.
A RECTILINEAL figure is said to be inscribed in an- Book IV.

other rectilineal figure, when all the angles of the inscribed
figure are upon the sides of the figure
in which it is inscribed, each upon
each.

II.
In like manner, a figure is said to be de-

scribed about another figure, when all
the sides 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 inscrib-

ed in a circle, when all the angles of
the inscribed figure are upon the cir-
cumference of the circle.

Book IV.

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.

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v.
In like manner, a circle is said to be in-

scribed in a rectilineal figure, when
the circumference of the circle touches
each side of the figure.

VI.
A circle is said to be described about a

rectilineal figure, when the circumfe-
rence of the circle passes through all
the angular points of the figure about
which it is described.

VII.
A straight line is said to be placed in a circle, when the ex-

tremities of it are in the circumference of the circle.

PROP. I. PROB.

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

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