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THE ELEMENTS OF PLANE GEOMETRY.

BOOK III.

THE CIRCLE.

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SECTION I.

ELEMENTARY PROPERTIES.

DEF. 1. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle.

DEF. 2. A radius of a circle is a straight line drawn from the centre to the circumference,

DEF. 3.

A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

THEOR. I. The distance of a point from the centre of a circle is less than, equal to, or greater than the radius, according as the point is within, on, or without the circum. ference.

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then shall OP be less than, equal to, or greater than the radius, according as P is within, on, or without the circumference.

The straight line passing through O and P will meet the circumference in two points Q, Q', and in no other points, since there are only two points on the line whose distances from O are equal to the radius.

If P is between Q and Q', P is within the circumference, and OP is less than OQ, that is, less than the radius.

If P coincide with or Q', P is on the circumference, and OP is equal to OQ or OQ”, that is, equal to the radius.

If P is on OQ or OQ produced, P is without the circumference, and OP is greater than OQ or OQ', that is, greater than the radius.

Q.E.D.

COR. A point is within, on, or without the circumference of a circle, according as its distance from the centre is less than, equal to, or greater than the radius.

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