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E L E M E N T S

OF

G E O M E T R Y.

Β Ο Ο Κ ΙΙΙ. .

DEFINITIONS.

TH

A. *HE radius of a circle is the straight line drawn from Book III. the centre to the circumference.

1. A straight line is said to

touch a circle, when it
meets the circle, and be-
ing produced does not cut
it.

II.
Circles are said to touch one

another, which meet, but
do not cut one another.

III.
Straight lines are said to be equally di-

stant from the centre of a circle,
when the perpendiculars drawn to
them from the centre are equal.

IV.
And the straight line on which the

greater perpendicular falls, is said to
be farther from the centre.

B.

F 3

Book III.

B.
An arch of a circle is any part of the circumference.

V.
A segment of a circle is the figure con-

tained by a straight line, and the
arch which it cuts off.

VI.
An angle in a segment is the angle

contained by two straight lines
drawn from any point in the cir-
cumference of the segment, to the
extremities of the straight line
which is the base of the segment.

VII.
And an angle is said to infift or ftand

upon the arch intercepted between
the straight lines that contain the
angle.

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Let ABC be the given circle ; it is required to find its centre.

Draw

a 10. 1.

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· Draw within it any straight line AB, and bisect a it in D; Book III. from the point D drawb DC at right angles to AB, and produce it to E, and bisect CE in F: The point F is the centre d u. i. of the circle ABC.

For, if it be not, let, if pollible, G be the centre, and join GA, GD, GB: Then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two fides AD, DG are equal to the two BD, DG, each to each ; and the base GA is equal to the base GB, because they are radii of the same circle : therefore the angle ADG is equal c

c 8. 1. to the angle GDB : But when a straight line standing upon another straight line makes the adjacent angles equal to one another, each

В of the angles is a right angle d.

d 7. def. s. Therefore the angle GDB is a right

I angle : But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impossible : Therefore G is not the centre of the circle ABC: In the same manner, it can be shown, that no other point but F is the centre: that is, F is the centre of the circle ABC: Which was to be found.

NG

Cor. From this it is manifest that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bifects the other,

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F any two points be taken in the circumference of

a circle, the straight line which joins them Thall fall within the circle.

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Book III.

Let ABC be a circle, and A, B any two points in the ciscumference; the straight line drawn

с
from A to B shall fall within the
circle.

Take any point in AB as E; find
D the centre of the circle ABC;

D
join AD, DB and DE, and let' DE
meet the circumference in F. Then
because DA is equal to DB, the

A

B В a s. I. angle DAB is equal to the angle

F
DBA; and because AE, a side of

the triangle DAE, is produced to B, the angle DEB is b 16. I, greater b than the angle DAE ; but DAE is equal to the

angle DBE ; therefore the angle DEB is greater than the

angle DBE: Now to the greater angle the greater fide is opc 19. 1. polited; DB is therefore greater than DE: but BD is equal

to DF; wherefore DF is greater than DE, and the point E is therefore within the circle. The same may be demonstrated of any other point between A and B, therefore AB is within the circle. Wherefore, if any two points, , &c. Q. E. D.'

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F a straight line drawn through the centre of a

circle bitect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles ; and, if it cut it at right angles, it will bifect it.

Let ABC be a circle, and let CD, a straight line drawn through the centre, bisect any straight line AB, which does not pass through the centre, in the point F: It cuts it also at right angles.

Take a E the centre of the circle, and join EA, EB. Then, because AF is equal to FB, and FE common to the two

triangles

a 1. 3.

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