1. DEFINITIONS. A circle is a portion of a plane bounded by a curve, all the points of which are equally distant from a point within it called the centre. The curve which bounds the circle is called its circumference. Any straight line drawn from the centre to the circumference is called a radius. Any straight line drawn through the centre. and terminated each way by the circumference is called a diameter. In the figure, O is the centre, and the curve ABCEA is the circumference of the circle; the circle is the space included within the circumference; OA, OB, OC, are radii; AOC is a diameter. By the definition of a circle, all its radii are equal; also all its diameters are equal, each being double the radius. If one extremity, O, of a line OA is fixed, while the line revolves in a plane, the other extremity, A, will describe a circumference, whose radii are all equal to OA. 2. Definitions. An are of a circle is any portion of its circumfer ence; as DEF. A chord is any straight line joining two points of the circumference; as DF. The arc DEF is said to be subtended by its chord DF. Every chord subtends two arcs, which together make up the whole circumference. Thus DF subtends both the arc DEF and the arc DCBAF. Wher an arc and its chord are spoken of, the arc less than a semi-circumference, as DEF, is always understood, unless otherwise stated. A segment is a portion of the circle included between an arc and its chord; thus, by the segment DEF is meant the space included between the arc DF and its chord. A sector is the space included between an arc and the two radii drawn to its extremities; as A OB. 3. From the definition of a circle it follows that every point within the circle is at a distance from the centre which is less than the radius; and every point without the circle is at a distance from the centre which is greater than the radius. Hence (I. 40), the locus of all the points in a plane which are at a given distance from a given point is the circumference of a circle described with the given point as a centre and with the given distance as a radius. 4. It is also a consequence of the definition of a circle, that two circles are equal when the radius of one is equal to the radius of the other, or when (as we usually say) they have the same radius. For if one circle be superposed upon the other so that their centres coincide, their circumferences will coincide, since all the points of both are at the same distance from the centre. If when superposed the second circle is made to turn upon its centre as upon a pivot, it must continue to coincide with the first. 5. Postulate. A circumference may be described with any point as a centre and any distance as a radius. ARCS AND CHORDS. PROPOSITION I.-THEOREM. 6. A straight line cannot intersect a circumference in more than two points. For, if it could intersect it in three points, the three radii drawn to these three points would be three equal straight lines drawn from the same point to the same straight line, which is impossible (I. 36). 5* PROPOSITION II.-THEOREM. 7. Every diameter bisects the circle and its circumference. Let AMBN be a circle whose centre is 0; then, any diameter AOB bisects the circle and its circumference. M N B For, if the figure ANB be turned about AB as an axis and superposed upon the figure AMB, the curve ANB will coincide with the curve AMB, since all the points of both are equally distant from the centre. The two figures then coincide throughout, and are therefore equal in all respects. Therefore, AB divides both the circle and its circumference into equal parts. 8. Definitions. A segment equal to one half the circle, as the segment AMB, is called a semi-circle. An arc equal to half a circumference, as the arc AMB, is called a semi-circumference. PROPOSITION III.-THEOREM. 9. A diameter is greater than any other chord. Let AC be any chord which is not a diameter, and AOB a diameter drawn through A: then AB > AC. For, join OC. Then, AO+ OC > AC (I. 66); that is, since all the radii are equal, AO+ OB>AC, or AB > AC. C B PROPOSITION IV.-THEOREM. 10. In equal circles, or in the same circle, equal angles at the centre intercept equal arcs on the circumference, and conversely. Let O, O', be the centre of equal circles, and AOB, A' O'B', equal angles at these centres; then, the intercepted arcs, AB, A'B', are equal. For, one of the angles, together with its arc, may be superposed upon the other; and when the equal angles coincide, their intercepted arcs will evidently coincide also. Conversely, if the arcs AB, A'B' are equal, the angles AOB, A'O'B' are equal. For, when one of the arcs is superposed upon its equal, the corresponding angles at the centre will evidently coincide. If the angles are in the same circle, the demonstration is similar. 11 Definition. A fourth part of a circumference is called a quadrant. It is evident from the preceding theorem that a right angle at the centre intercepts a quadrant on the circumference. Thus, two perpendicular diameters, AOC, BOD, divide the circumference into four quadrants, AB, BC, CD, DA. B D A PROPOSITION V.-THEOREM. 12. In equal circles, or in the same circle, equal arcs are subtended by equal chords, and conversely. Let O, O', be the centres of equal circles, and AB, A'B', equal arcs; then, the chords AB, A'B', are equal. For, drawing the radhi to the extremities of the arcs, the angles O and O' are equal (10), and consequently the triangles AOB, A'O'B', are equal (I. 76). Therefore, AB A'B'. = Α B A' B' Conversely, if the chords AB, A'B', are equal, the triangles AOB, A'O'B' are equal (I. 80), and the angles O, O' are equal. Therefore (10), arc AB =arc A'B'. If the arcs are in the same circle, the demonstration is similar. PROPOSITION VI.-THEOREM. 13. In equal circles, or in the same circle, the greater arc is subtended by the greater chord, and conversely; the arcs being both less than a semi-circumference. Let the arc AC be greater than the arc AB; then, the chord AC is greater than the chord AB. M M B C For, draw the radii OA, OB, OC. In the triangles AOC, A OB, the angle AOC is obviously greater than the angle AOB; therefore, (I. 84), chord AC chord AB. Conversely, if chord AC chord AB, then, arc AC > arc AB. For, in the triangles AOC, AOB, the side AC> the side AB; therefore (I. 85), angle AOC > angle AOB; and consequently, arc AC are AB. 14. Scholium. If the arcs are greater than a semi-circumference, the contrary is true; that is, the arc AMB, which is greater than the arc AMC, is subtended by the less chord; and conversely. PROPOSITION VII.-THEOREM. 15. The diameter perpendicular to a chord bisects the chord and the arcs subtended by it. Let the diameter DOD' be perpendicular to the chord AB at C; then, 1st, it bisects the chord. For, the radii OA, OB being equal oblique lines from the point O to the line AB, cut off equal distances from the foot of the perpendicular (I. 36); therefore, ACBC. D' 2d. The subtended arcs ADB, AD'B, are bisected at D and D', respectively. For, every point in the perpendicular DOD' drawn at the middle of AB being equally distant from its extremities A and B (I. 38), the chords AD and BD are equal; therefore, (12), the arcs AD and BD are equal. For the same reason, the arcs AD' and BD' are equal. 16. Corollary I. The perpendicular erected upon the middle of a chord passes through the centre of the circle, and through the middle of the arc subtended by the chord. Also, the straight line drawn through any two of the three points O, C, D, passes through the third and is perpendicular to the chord AB. B |