BOOK III. THE CIRCLE. CHAPTER I. GENERAL PROPERTIES OF THE CIRCLE. Definitions. 194. Def. The circumference of a circle is the total length of the· curve-line which forms it. 195. Def. An arc of a circle is a part of the curve which forms it. Minor conjugate arc. 196. Def. When two arcs together make an entire circle, they are called conjugate arcs, and the one is said to be the conjugate of the other. Major conjugate arc. 197. Def. When two conjugate arcs are equal, each of them is called a semicircle. 198. Def. When two conjugate arcs are unequal, the lesser is called the minor arc, and the greater the major arc. 199. Def. A chord is a straight line between two points of a circle. 200. Def. A secant is a straight line which intersects a circle. REMARK. A secant may be considered as a chord with one or both of its ends produced, and a chord as that part of a secant contained within the circle. 201. Def. The diameter of a circle is a chord which passes through its centre. 202. Def. A segment of a circle is composed of a chord and either of the arcs between its extremities. 203. Def. A sector is formed of two radii and the arc included between them. To a pair of radii may belong either of the two conjugate arcs into which their ends divide the circle. 204. Def. Concentric circles are those which have the same centre. 205. Def. A tangent to a circle is any straight line which touches the circle without intersecting it. 206. Special Axioms relating to the Circle. I. A circle has only one centre. II. Every point at a distance from the centre less than the radius is within the circle. III. Every point at a distance from the centre equal to the radius is on the circle. IV. Every point at a distance from the centre greater than the radius is without the circle. THEOREM I. 207. Circles of equal radii are identically equal. Hypothesis. Two circles of which O and P are centres, and Radius OQ = radius PR. Conclusion. The circles are identically equal. Proof. Apply the one circle to the other in such manner that the centre O shall coincide with P, and OQ with PR. Then 1. Because 0Q= PR, Pointpoint R. 2. Because each point of the one circle is at the R P distance OQ from the centre, it will fall on the other circle. (§ 206, Ax. III.) Therefore the circles are identically equal. Q.E.D. THEOREM II. 208. Equal arcs of equal circles are identically equal, subtend equal angles at centre, and contain equal chords. Hypothesis. AB, CD, equal arcs around the centres O and P. OA OB = PC = PD. Conclusion. The an gles AOB and CPD and the chords AB and CD are equal. B D Proof. Apply the sector OAB to the sector PCD so that the centre O shall fall on P, and the radius OA on the radius PC. Then— 2. Because the radii are all equal, every part of the arc AB will fall on some part of the circle to which the arc CD belongs (§ 206, III.). THEOREM III. 209. Equal angles between radii include equal arcs on the circle and equal chords. Hypothesis. OA, OB, OP, OQ, four radii of a circle such that 4. Because all the radii are equal, the arcs will coincide between P and Q. Therefore the arcs are identically equal. Q.E.D. 210. Corollary. Sectors of equal angles in equal circles are identically equal, and every line in the one sector is identically equal to the corresponding line in the other (§ 174). Lemma. 211. A sum of two arcs at the centre subtends an angle. equal to the sum of the angles which each arc subtends separately. Proof. The arc AB subtends the angle AOB, the arc BC subtends the angle BOC, and the arc ABC subtends the angle AOC. But AOC is by definition the sum of the B angles, and ABC is the sum of the arcs; which proves the lemma. The Measurement of Angles by means of Arcs. 212. From the preceding theorems it follows that to every arc of given length in a given circle corresponds a definite angle, and to every angle corresponds a definite length of arc. To express corresponding arcs and angles in the shortest way, we call the arc corresponding to the angle AOB the arc angle AOB, and we call the angle corresponding to an arc the angle arc. Thus, in the following figure, Angle AOB = angle arc AB. = Angle AOC angle arc ABC. Combining Theorem III. with the above lemma, it follows that arcs can be taken as the measure of the corresponding angles, and vice versa. In the figure the circle is divided into eight sectors, and since 360° 45°, each of these sectors subtends = E D 45°; arc AB B H Angle 40G 270°; arc ABCDEFG Angle AOH= 315°; arc ABCDEFGH = 270°. = 315°. Angle 404 360°; arc ABCDEFGHA= 360°. 213. The following are the principles to which we are thus led: I. In the same circle or in equal circles, the greater arc measures the greater angle. II. A minor arc, or an arc less than a semicircle, measures an angle less than a straight angle. |