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Book II. Then the rectangle under BE, EF, together with the fquare of GE, are equal to the fquare of GF, or GH d; but the fquare of GH is equal to the fquares of GE, EH ; d Def. 15 therefore the rectangle under BE, EF, together with the fquare of GE, are equal to the fquares of HE, EG. Take the fquare of GE from both, and the rectangle under BE, EF, that is, BD, is equal to the fquare of EH. Wherefore, &c.

I.

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A right line is faid to touch a circle, when drawn to the famê, and being produced, does not cut the circle.

III.

Circles are faid to touch each other, which, meeting, do not cut one another.

IV.

Right lines in a circle are faid- to be equally distant from the center, when perpendiculars drawn from the center to each of them are equal, and that line upon which the greatest perpendicular falls is the leaft line.

Definition 19th, 1.

V.

'VI.

An angle of a fegment is the angle contained by the right line and circumference of the circle.

VII.

Book III.

An angle is faid to be in a fegment, when right lines are drawn from fome point in the circumference to the ends of that line which is the base of the fegment, which lines contain the angle.

VIII. But,

BOOK III.

a 10. I.

b II. I.

c def. 15. 1. d 8. r.

VIII.

But, when the right lines containing the angle do receive any part of the circumference, then the angle is faid to stand upon that circumference.

IX.

A fector of a circle is that figure which is contained by two right lines, drawn from the center, and the circumference between them.

X.

Similar fegments of circles are thofe which include equal angles or whereof the angles in them are equal.

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Required to find the centre of the given circle ABC.

Draw in it the line AB, which bifect in Da, by the right line CD at right angles to AB, and produce CD to E; bifect EC in F; which point is the centre of the circle ABC.

If not, let G be the centre; join GB, GA, GD; then, becaufe AD is equal to DB, DG is common, the base AG is equal to GB, and the angle ADG to GDBd; therefore, each of them is a right angle; but FDB is a right angle; therefore, GDB is equal to FDB, a part to the whole, which is impof€ Ax. 9. 1. fible¤; therefore, no point but F can be the centre. Wherefore, &c.

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Cor. Hence, if, in a circle, any right line cut another right line into two equal parts, the centre of the circle will be in that line which cuts the other into two equal parts.

PRO P. II. THE O R.

IF any two points be affumed in the circumference of a circle, the right line joining these points will fall within the circle.

Let the circle be ABC, A and B the points in its circumference, the right line AB, joining thefe points, will fall within the circle. Find D the centre of the circle"; join DA, DB, and draw DF, cutting the right line AB in the point E; then b def. 15. 1. the right lines DA, DB, DF, are equal. But DF is greater CAX. 9. 1. than DE; therefore, DA, DB, are likewife greater than DE; but DB, DA, reach the circumference; therefore DE does not

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