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ELEMENTS

OF

GEOMETRY.

BOOK III.

DEFINITIONS.

A.

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

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I.

A ftraight line is faid to

touch a circle, when it

meets the circle, and being produced does not cut it.

II.

Circles are faid to touch one another, which meet, but

do not cut one another.

III.

Straight lines are faid to be equally di-
ftant from the centre of a circle,
when the perpendiculars drawn to
them from the centre are equal.
IV.

And the ftraight line on which the
greater perpendicular falls, is faid to
be farther from the centre.

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

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

centre.

Draw

Draw within it any ftraight line AB, and bifect a it in D; from the point D draw b DC at right angles to AB, and produce it to E, and bisect CE in F: The point F is the centre of the circle ABC.

C

Book III.

a 10. 1.

b 11. 1.

c 8. 1.

F

For, if it be not, let, if poffible, G be the centre, and join GA, GD, GB: Then, becaufe 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, becaufe they are radii of the fame circle: therefore the angle ADG is equal to the angle GDB: But when a ftraight line ftanding upon another ftraight line makes the adjacent angles equal to one another, each of the angles is a right angle d. Therefore the angle GDB is a right angle: But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the lefs, which is impoffible: Therefore G is not the centre of the circle ABC: In the fame 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.

E

COR. From this it is manifeft that if in a circle a straight line bifect another at right angles, the centre of the circle is in the line which bifects the other.

d 7. def. 1.

PROP. II. THEOR.

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F any two points be taken in the circumference of a circle, the straight line which joins them fhall fall within the circle.

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Book III.

a 5. I.

C

Let ABC be a circle, and A, B any two points in the cit cumference; the ftraight line drawn from A to B fhall fall within the circle.

Take any point in AB as E; find D the centre of the circle ABC, join AD, DB and DE, and let DE meet the circumference in F. Then because DA is equal to DB, the angle DAB is equal to the angle DBA; and becaufe AE, a fide of

D

E

B

F

the triangle DAE, is produced to B, the angle DEB is b 16. 1. greater 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. pofited; 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 fame may be demonftrated 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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PROP. III. THEOR.

Fa ftraight line drawn through the centre of a circle bifect a ftraight 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.

a 1. 3.

Let ABC be a circle, and let CD, a straight line drawn through the centre, bifect any ftraight line AB, which does not país 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

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