8. An angle in a segment is one formed by two straight lines drawn from any point in the arc of the segment to the extremities of its chord. NOTE. (i) It will be shewn in Proposition 21, that all angles in the same segment of a circle are equal. NOTE. (ii) The angle of a segment (as distinct from the angle in a segment) is sometimes defined as that which is contained between the chord and the arc; but this definition is not required in any proposition of Euclid. 9. An angle at the circumference of a circle is one formed by straight lines drawn from a point on the circumference to the extremities of an arc: such an angle is said to stand upon the arc by which it is subtended. 11. A sector of a circle is a figure bounded by two radii and the arc intercepted between them. SYMBOLS AND ABBREVIATIONS. In addition to the symbols and abbreviations given on page 11, we shall use the following. for circle, ce for circumference. From D draw DC at right angles to AB; ce and produce DC to meet the O at E and C. I. 10. I. 11. I. 10. Proof. First, the centre of the circle must be in EC: for if not, suppose the centre to be at a point G outside EC. Because Join AG, DG, BG. Then in the ▲ GDA, GDB, DA = DB, and GD is common; Constr. and GA= GB, for by supposition they are radii; .. the GDA = the GDB; .. these angles, being adjacent, are rt. angles. But the ... the CDB is a rt. angle; I. 8. Constr. Ax. 11. the part equal to the whole, which is impossible. So it may be shewn that no point outside EC is the centre; the centre lies in EC. .. F, the middle point of the diameter EC, must be the centre of the ABC. Q.E.F. COROLLARY. The straight line which bisects a chord of a circle at right angles passes through the centre. If any two points are taken in the circumference of a circle, the chord which joins them falls within the circle. Let ABC be a circle, and A and B any two points on its Oce. Then shall the chord AB fall within the circle. ... the DEB is also greater than the DBE. .. in the ▲ DEB, the side DB, which is opposite the greater angle, is greater than DE which is opposite the less: I. 19. that is to say, DE is less than DB, a radius of the circle; .. E falls within the circle. Similarly, any other point between A and B may be shewn to fall within the circle. ... AB falls within the circle. Q.E.D. NOTE. A part of a curved line is said to be concave to a point, when for every chord (taken so as to lie between the point and the curve) all straight lines drawn from the given point to the intercepted arc are cut by the chord: if, when any chord whatever is taken, no straight line drawn from the given point to the intercepted arc is cut by the chord, the curve is said to be convex to that point. Proposition 2 proves that the whole circumference of a circle is concave to its centre. If a straight line drawn through the centre of a circle bisects a chord which does not pass through the centre, it shall cut the chord at right angles. Conversely, if it cuts the chord at right angles, it shall bisect it. B Let ABC be a circle; and let CD be a st. line drawn through the centre, and AB a chord which does not pass through the centre. Let CD bisect the chord AB at F. First. Construction. Proof. Because Find E the centre of the circle; and join EA, EB. Then in the ▲ AFE, BFE, AF = BF, and FE is common; and AE = BE, being radii of the circle; .. the AFE = the BFE ; .. these angles, being adjacent, are rt. angles; Conversely. Let CD cut the chord AB at rt. angles. Construction. Proof. III. 1. Hyp. I. 8. Q.E.D. Find E the centre; and join EA, EB. In the EAB, because EA EB, = I. Def. 15. I. 5. Proved. EFB, being rt. angles; Hyp. and EF is common; .'. AF = BF ; that is, CD bisects AB at F. I. 26. Q.E.D. PROPOSITION 4. THEOREM. If in a circle two chords cut one another, which do not both pass through the centre, they cannot both be bisected at their point of intersection. Let ABCD be a circle, and AC, BD two chords which intersect at E, but do not both pass through the centre. Then AC and BD shall not be both bisected at Ē. CASE I. If one chord passes through the centre, it is a diameter, and the centre is its middle point; ... it cannot be bisected by the other chord, which by hypothesis does not pass through the centre. CASE II. If neither chord passes through the centre; then, if possible, suppose E to be the middle point of both; that is, let AE = EC; and BE = ED. Construction. Find F, the centre of the circle. Join EF. III. 1. Proof. Because FE, which passes through the centre, bisects the chord AC, .. the FEC is a rt. angle. Hyp. III. 3. And because FE, which passes through the centre, bisects the chord BD, ..the FED is a rt. angle. the whole equal to its part, which is impossible. Hyp. III. 3. .. AC and BD are not both bisected at E. Q.E.D. |