THEOREM XV. 170. Like powers, or like roots, of the terms of a proportion form a proportion. 171. SCHOLIUM. The product of two quantities implies that at least one is numerical. In (169) and (170), all the quantities must be numerical. In (160) and (168), all the quantities must be of the same kind. BOOK III. THE CIRCLE. DEFINITIONS. 172. A circle is a plane bounded by a curve, all the points of which are equally distant from a point within, called the centre. The Circumference of a circle is the curve which bounds it. A C B An Arc is a part of the circumference; as, AC. A. Semicircumference is an arc equal to half of the circumference. A Radius is a straight line extending from the centre to any point in the circumference; as, OC. 173. A Diameter of a circle is a straight line passing through the centre and terminating each way in the circumference; as, АВ. 174. A Chord is a straight line joining any two points in the circumference; as, P ED. The arc EPD is said to be subtended by its chord ED. Every chord subtends two E arcs, whose sum equals the whole circumference. Whenever an arc and its chord are spoken of, the less arc is meant. 175. A Segment of a circle is the portion enclosed by an arc and its chord; as, FGH. A Semicircle is a segment equal to one-half of the circle. F G 176. A Sector of a circle is the por- H tion enclosed by an arc and the radii drawn to its extremities; as, OFM. 177. A Tangent is a straight line which touches the circumference but does not intersect it; as, ABC. The common point B is called the point of contact, or the point of tangency. D M 178. A Secant is a straight line which cuts the circumference in two points; as, DE. 179. An Inscribed Angle is one whose vertex is in the circumference and whose sides are chords; as, ABE. 180. An Inscribed Polygon is one whose sides are chords of a circle; as, ABCDEF. The circle is then said to be circumscribed E D diameters are equal. It also follows from the definition that circles are equal when their radii are equal. CHORDS, ARCS, ANGLES AT THE CENTRE, SECANTS, AND RADII. THEOREM I. 183. Any diameter bisects the circle and its circumference. Let ABCD be a O, and AB any diameter. On AB as an axis, revolve the portion ABC till it falls in the plane of ABD. Then the curve ACB coincides with the curve ADB, for all the points in each are equally distant from the centre 0. |