surface, the moving point describes a physical circle. The limit of this physical circle, when the curved line has its thickness diminished endlessly, is the geometric circle. Def. 2.-The fixed point is the centre of the circle, and the distance between the fixed and moveable points is the radius of the circle. The curve itself, and especially where its length is under consideration, is commonly called the circumference of the circle. The symbol employed for the circle is O. 93° From the definitions of 92° we deduce the following corollaries: 1. All the radii of a O are equal to one another. 2. The is a closed figure; so that to pass from a point within the figure to a point without it, or vice versa, it is necessary to cross the curve. 3. A point is within the O, on the O, or without the O, according as its distance from the centre is less, equal to, or greater than the radius. 4. Two Os which have equal radii are congruent; for, if the centres coincide, the figures coincide throughout and form virtually but one figure. Def.-Circles which have their centres coincident are called concentric circles. 94°. Theorem.-A line can cut a circle in two points, and in two points only. Proof. Since the O is a closed curve (93°, 2), a line which cuts it must lie partly within the O and partly without. And the generating point (69°) of the line must cross the in passing from without to within, and again in passing from within to without. .. a line cuts a at least twice if it cuts the at all. Again, since all radii of the same are equal, if a line could cut a three times, three equal segments could be drawn from a given point, the centre of the O, to a given line. And this is impossible (63°, 3). Therefore a line can cut a O only twice. q.e.d. Cor. 1. Three points on the same circle cannot be in line; or, a circle cannot pass through three points which are in line. 95°. Def. 1.—A line which cuts a ○ is a secant or secant-line. Def. 2.-The segment of a secant included within the is a chord. Thus the line L, or AB, is a secant, and the segment AB is a chord. (21°) The term chord whenever involv A B L M о D ing the idea of length means the segment having its endpoints on the circle. But sometimes, when length is not involved, it is used to denote the whole secant of which it properly forms a part. Def. 3.-A secant which passes through the centre is a centre-line, and its chord is a diameter. Where length is not implied, the term diameter is sometimes used to denote the centre-line of which it properly forms a part. Thus M is a centre-line and CD is a diameter. 96°. Theorem.-Through any three points not in line I. One circle can be made to pass. 2. Only one circle can be made to pass. Proof.-Let A, B, C be three points. not in line. Join AB and BC, and let L and M be the A right bisectors of AB and BC respectively. 1. Then, because AB and BC intersect at B, M D B (79°, Cor.) and .. the O is equidistant from A, B, and C. (86°) with centre at O, and radius equal to OA, passes through B and C. q.e.d. 2. Any through A, B, and C must have its centre equally distant from these three points. But O is the only point in the plane equidistant from A, B, and C. (86°, Cor.) And we cannot have two separate s having the same centre and the same radius. (93°, 4) .'. only one circle can pass through A, B, and C. q.e.d. Cor. 1. Circles which coincide in three points coincide altogether and form one circle. Cor. 2. A point from which more than two equal segments can be drawn to a circle is the centre of that circle. Cor. 3. Since L is a centre-line and is also the right bisector of AB, .. the right bisector of a chord is a centre line. Cor. 4. The ▲AOB is isosceles, since OA=OB. Then, if D be the middle of AB, OD is a median to the base AB and is the right bisector of AB. (55°, Cor. 2) .. a centre-line which bisects a chord is perpendicular to the chord. Cor. 5. From Cor. 4 by the Rule of Identity, A centre line which is perpendicular to a chord bisects the chord. .. the right bisector of a chord, the centre-line bisecting the chord, and the centre-line perpendicular to the chord are one and the same. 97°. From 92°, Def., a circle is given when the position of its centre and the length of its radius are given. And, from 96°, a circle is given when any three points on it are given. It will be seen hereafter that a circle is determined by three points even when two of them become coincident, and in higher geometry it is shown that three points determine a circle, under certain circumstances, when all three of the points become coincident. Def.-Any number of points so situated that a circle can pass through them are said to be concyclic, and a rectilinear figure (14°, Def.) having its vertices concyclic is said to be inscribed in the circle which passes through its vertices, and the circle is said to circumscribe the figure. Hence the circle which passes through three given points is the circumcircle of the triangle having these points as vertices, and the centre of that circle is the circumcentre of the triangle, and its radius is the circumradius of the triangle. (86°, Def.) A like nomenclature applies to any rectilinear figure having its vertices concyclic. 98°. Theorem.-If two chords bisect one another they are both diameters. If AP=PD and CP=PB, then P is the centre. A C B D Proof. Since P is the middle point of both AD and CB (hyp.), therefore the right bisectors of AD and CB both pass through P. But these right bisectors also pass through the centre; (96°, Cor. 3) .. P is the centre. (24°, Cor. 3) q.c.d. 99°. Theorem.-Equal chords are equally distant from the centre; and, conversely, chords equally distant from the centre are equal. If ABCD and OE and OF are the perpendiculars from the centre upon these chords, then OE=OF; and conversely, if OE=OF, then ABCD. Proof. Since OE and OF are centre lines . in the As OBE and ODF OB=OD, EB=FD, A C E B F to AB and CD, (96°, Cor. 5) and they are right-angled opposite equal sides, Conversely, by the Rule of Identity, if OE=OF, then AB-CD. G q.e.d. 100°. Theorem.-Two secants which make equal chords LAPO=LCPO. q.e.d. Cor. i. . E and F are the middle points of AB and CD, (96°, Cor. 5) .. PE=PF, PA=PC, and PB=PD. Hence, secants which make equal chords make two pairs of equal line-segments between their point of intersection and the circle. Cor. 2. From any point two equal line-segments can be drawn to a circle, and these make equal angles with the centre-line through the point. 101°. As all circles have the same form, two circles which have equal radii are equal and congruent (93°, 4), (51°). Hence equal and congruent are equivalent terms when applied to the circle. Def. 1.-Any part of a circle is an arc. The word equal when applied to arcs means congruence or capability of superposition. Equal arcs come from the same circle or from equal circles. |