BOOK II. THE CIRCLE. DEFINITIONS. 127. A Circle is a plane figure bounded by a curve, all points of which are equally distant from a point within called the Centre. The curve which bounds the circle is called the Circumference, and any portion of it is called an Arc. 128. A Chord is a straight line which joins any two points on the circumference, as BC. A B D When a chord passes through the centre, it has its greatest length, and is called the Diameter. 129. A Radius is a straight line drawn from the centre to the circumference, and since, by definition, this distance is the same for the same circle, all radii are equal; and each radius is one-half of the diameter. 130. An arc equal to one-half the circumference is called a Semi-circumference, and an are equal to one-fourth of the cir cumference is called a Quadrant. 131. Two circles are Equal when they have equal radii, for they can evidently be applied one to the other so as to coincide throughout. 132. Two circles are Concentric when they have the same centre. 133. POSTULATE: the circumference of a circle can be described about any point as a centre and with any distance for a radius. 134. A Segment of a circle is a portion of a circle enclosed by an arc and its chord, as AMB, Fig. 1. 135. A Semicircle is a segment equal to one-half the circle, as ADC, Fig. 1. 136. A Sector of a circle is a portion of the circle enclosed by two radii and the arc which they intercept, as ACB, Fig. 2. 137. A Tangent is a straight line which touches the circumference, but does not intercept it, however far produced. The point in which the tangent touches the circumference is called the Point of Contact, or Point of Tangency. 138. Two Circumferences are tangent to each other when they are tangent to a straight line at the same point. 139. A Secant is a straight line which intersects the circumference in two points, as AD, Fig 3. 140. A straight line is Inscribed in a circle when its extremities lie in the circumference of the circle, as AB, Fig. 1. An angle is inscribed in a circle when its vertex is in the circumference, and its sides are chords of that circumference, as ABC, Fig. 1. A polygon is inscribed in a circle when its sides are chords of the circle, as ABC, Fig. 1. A circle is inscribed in a polygon when the circumference touches the sides of the polygon but does not intersect them, as in Fig. 4. 141. A polygon is Circumscribed about a circle when all the sides of the polygon are tangents to the circle, as in Fig. 4. A circle is circumscribed about a polygon when the circumference passes through all the vertices of the polygon, as in Fig. 1. M 142. Every diameter bisects the circle and its circumference. For if we fold over the segment AMB on AB as an axis until it comes into the plane of APB, the arc AMB will coincide with the arc APB; because every point in each is equally distant from the centre 0. 143. A straight line cannot meet the circumference of a circle in more than two A P B points. For if it could meet it in three points, these three points would be equally distant from the centre (127). There would then be three equal straight lines drawn from the same point to the same straight line, which is impossible (58 a). PROPOSITION I. THEOREM. 144. In equal circles, or in the same circle, equal arcs are intercepted by equal central angles and have equal chords. Let ABM and A'B'M' be two equal circles, in which 40=ZC. To prove that the arc AB: arc A'B' and the chord AB = chord A'B'. 1. Place the circle ABM upon A'B'M' so that their centres may coincide, and A fall upon A'; then, since they are equal circles, they will coincide throughout. Since CZ C', the radius CB will take the direction of C'B', and, being radii of equal circles, B will fall upon B'. Therefore the arc AB will coincide with arc A'B' and be equal to it. Q.E.D. 2. The two triangles ACB and A'C'B' have AC= A'C' and BC= B'C', being radii of equal circles (by 130), and ≤ C = /C" by hypothesis. Therefore (by 86) the two triangles are equal in all their parts, or AB = A'B'. Q.E.D. 145. COR. 1. The converse is true, that is: In equal circles, or in the same circle, equal central angles intercept equal arcs on the circumference. COR. 2. Also the converse: In equal circles, or in the same circle, equal chords subtend equal arcs and equal angles at the centre. PROPOSITION II. THEOREM. 146. The diameter perpendicular to a chord bisects the chord and its subtended arcs. In the circle ADB, let the diameter CD be perpendicular to the chord AB. To prove that DC bisects AB and its subtended arcs. Let O be the centre of the circle, and join OA and OB. Then since OA = OB, the triangle OAB is isosceles; and the line CD, passing through the vertex perpendicular to the base, bisects the base and also the vertical angle (94). Hence AOC = Z BOC, and arc AC = are BC (144). Subtracting the equal arcs AC and BC from the semicircumferences CAD and CBD, we have arc AD =arc BD. Therefore the diameter bisects the chord AB and its subtended arcs ACB and ADB. 147. COR. The perpendicular erected at the middle point of a chord passes through the centre of the circle. And in general, if a straight line is drawn so as to satisfy any two of the following conditions: 1. Passing through the centre, 2. Perpendicular to the chord, 4. Bisecting the less subtended arc, 5. Bisecting the greater subtended arc, it will also satisfy the remaining conditions. |