THE ELEMENTS OF PLANE GEOMETRY. BOOK III. THE CIRCLE. 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 circumference. Let O be the centre of the circle, P any other point: 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 Q 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. |