THE ELEMENTS OF PLANE GEOMETRY. BOOK III. THE CIRCLE. SECTION I. ELEMENTARY PROPERTIES. DEF. 1. A circle is a plane figure contained by one iine, 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. 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. 1 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. Two points in a straight line equidistant from a point O`in it, and on opposite sides of O, are said to be symmetrical with respect to the point O. When to every point in a figure there corresponds another point of it symmetrical with the first with respect to a certain point O, the figure is said to be symmetrical or to have pointsymmetry with respect to O, which is then called its centre of symmetry, or simply its centre. Hence a circle is symmetrical with respect to its centre; or, the centre of a circle is a centre of symmetry. Ex. I. Ex. 2. Shew that a circle has only one centre. Shew that every parallelogram has point-symmetry with respect to the intersection of its diagonals. Ex. 3. If a quadrilateral has point-symmetry with respect to the intersection of its diagonals, it must be a parallelogram. THEOR. 2. Any diameter of a circle divides it into two identically equal parts, called semicircles. Let O be the centre, and AOB any diameter of the circle APBQ, dividing it into two parts APB, AQB: Ө then shall APB be identically equal to AQB. |