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its Part between the Convexity of the Circle and the assum'd Point, will be equal to the Square of the Tangent Line; which was to be demonstrated.

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17 of ebis.

THEOREM.
If some Point be taken without a Circle, and two

Right Lines be drawn from it to the Circle, so
that one cuts it, and the other falls upon it;
and if tbe Rettangle under the whole Secant Line,
and the Part thereof, without the Circle, be
equal to the Square of the Line falling upon the
Circle, then this last Line will touch the Circle.
ET fome Point D be assumed without the Circle

ABC, and from it draw two Right Lines DCA, DB, to the Circle in such Manner that DCA cuts the Circle, and D B falls upon it: And let the Rectangle under AD, DC, be equal to the Square of DB. I say, the Right Line D B touches the Circle.

For let the Right Line DE be drawn * touching the Circle ABC, and find F the Center of the Circle,

and join EF, FB, FD. † 18 of this. Then the Angle FED is ta Right Angle. And

because DE touches the Circle ABC, and DCA cuts it, the Rectangle under AD, and DC, will be

equal to the Square of DE. But the Rectangle un1 By Hyp. der AD and DC, is equal to the Square of DB.

Wherefore the Square of D E shall be equal to the

of DB. And so the Line DE will be equal to the Line D B. But EF is equal to FB: Therefore the two Sides DE, EF, are equal to the two Sides DB, BF; and the Base FD is common. Wherefore the Angle DEF is equal to the Angle DBF; but DEF is a Right Angle; wherefore DBF is also a Right Angle, and FB produced is a Diameter. But a Right Line drawn at Right Angles, on the End of the Diameter of a Circle, touches the Circle; therefore BD necessarily touches the Circle, We prove this in the same Manner, if the Center of the Circle be in the Right Line CA. If therefore any Point be asumed without a Circle, and two Right

Lines

Square

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Lines be drawn from it to the Circle, so that one cuts it, and the other falls upon it; and if the Rectangle under the whole Secant Line, and the Part thereof, without the Circle, be equal to the Square of the Line falling upon the Circle; then this last Line will touch the Circle; which was to be demonstrated:

Coroll. Hence, if from any Point without a Circle,

several Right Lines AB, AC, are drawn cutting the Circle ; thé Rectangles comprehended under the whole Liñes AB, AC, and their external Parts AE, AF, are equal between themselves. For if the Tangent AD be drawn, the Rectangle under BA and AE, is equal to the Square of AD; and the Rectangle under CA and AF, is equal to the same Square of AD: Therefore the Rectangles thall be equal.

The End of the THIRD BOOK,

H

EUCLID:

100

E U CL I D's

ELEMENTS.

B:0 O K IV.

DEFINITIONS.

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I. Right-lined Figure is said to be inscribed

in a Right-lined Figure, when every one
of the Angles of the inscribed Figure,

touches every one of the sides of the Fi-
gure, wherein it is described.
II. In like Manner a Figure is said to be described

about a Figure, when every one of the Sides of
the Figure, circumscribed, touches every one of
the Angles of the Figure about which it is cir-

cumscribed.
III. Å Right-lined Figure is said to be inscribed in

a Circle, when every one of the Angles of that
Figure which is inscribed, touches the Circumfe-

rence of the Circle.
IV. A Right-lined Figure is said to be described a-

bout a Circle, when every one of the Sides of the
circumscribed Figure, touches the Circumference
of the Circle.

V. So

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