bisecting AB, which does not pass through the centre, cuts AB at right angles. Again, let CD cut AB at right angles; CD also bisects AB, that is, AF is equal to FB. The same construction being made, because the radii EA, FB are equal to one another, the angle EAF is equal (5. 1.) to the angle EBF; and the right angle AFE is equal to the right angle BFE: Therefore, in the two triangles EAF, EBF, there are two angles in one equal to two angles in the other; now the side EF, which is opposite to one of the equal angles in each, is common to both; therefore the other sides are equal to (26. 1.): AF therefore is equal to FB. COR. 1. Hence, the perpendicular through the middle of a chord, passes through the centre; for this perpendicular is the same as the one let fall* from the centre on the same chord, since both of them passes through the middle of the chord. COR. 2. It likewise follows, that the perpendicular drawn through the middle of a chord, and terminated both ways by the circumference of the circle, is a diameter, and the middle point of that diameter is therefore the centre of the circle. PROP. IV. THEOR. If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other. Let ABCD be a circle, and AC, BD two straight lines in it, which cut one another in the point E, and do not both pass through the centre: AC, BD do not bisect one another. F D E B For if it is possible, let AE be equal to EC, and BE to ED; if one of the lines pass through the centre, it is plain that it cannot be bisected by the other, which does not pass through the centre. But if neither of them pass through the centre, take (1. 3.) F the centre A of the circle, and join EF: and because FE, a straight line through the centre, bisects another AC, which does not pass through the centre, it must cut it at right (3. 3.) angles; wherefore FEA is a right angle. Again, because the straight line FE bisects the straight line BD, which does not pass through the centre, it must cut it at right (3. 3.) angles; wherefore FEB is a right angle and FEA was shown to be a right angle: therefore FEA is equal to the angle FEB, the less to the greater, which is impossible; therefore AC, BD, do not bisect one another. PROP. V. THEOR. If two circles cut one another, they cannot have the same centre. Let the two circles ABC, CDG cut one another in the points B, C; they have not the same centre. 1 For, if it be possible, let E be their centre join EC, and draw any straight line EFG meeting the circles in F and G: and because E is the centre of the circle ABC, CE is equal to EF: Again, because E is the centre of the circle CDG, CE is equal to EG: but CE was shown to be equal to EF, therefore EF is equal to EG, the less to the greater, which is impossible: therefore E is not the centre of the circles, ABC, CDG. If two circles touch one another internally, they cannot have the same centre. Let the two circles ABC, CDE, touch one another internally in the point C; they have not the same centre. For, if they have, let it be F; join FC, and draw any straight line FEB meeting the circles in E and B; and because F is the centre of the circle ABC, CF is equal to FB; also, because F is the centre of the circle CDE, CF is equal to FE but CF was shown to be equal to FB; therefore FE is equal to FB, the less to the greater, which is impossible; Wherefore F is not the centre of the circles ABC, CDE. PROP. VII. THEOR. C F E B D If any point be taken in the diameter of a circle which is not the centre, of all the straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is, and the other part of that diameter is the least; and, of any others, that which is nearer to the line passing through the centre is always greater than one more remote from it; And from the same point there can be drawn only two straight lines that are equal to one another, one upon each side of the shortest line. Let ABCD be a circle, and AD its diameter, in which let any point F be taken which is not the centre: let the centre be E; of all the straight lines FB, FC, FG, &c. that can be drawn from F to the circumference, FA is the greatest; and FD, the other part of the diameter AD, is the least; and of the others, FB is greater than FC, and FC than FG. Join BE, CE, GE; and because two sides of a triangle are greater (20. 1.) than the third, BE, EF are greater than BF; but AE is equal to EB; therefore AE and EF, that is, AF, is greater than BF: Again, because BE is equal to CE, and FE common to the triangles BEF, CEF, C B K H D the two sides BE, EF are equal to the two CE, EF; but the angle BEF is greater than the angle CEF; therefore the base BF is greater (24 1.) than the base FC; for the same reason, CF is greater than GF. Again, because GF, FE are greater (20. 1.) than EG, and EG is equal to ED; GF, FE are greater than ED; take away the common part FE, and the remainder GF is greater than the remainder FD: therefore FA is the greatest, and FD the least of all the straight lines from F to the circumference; and BF is greater than CF, and CF than GF. Also there can be drawn only two equal straight lines from the point F to the circumference, one upon each side of the shortest line FD at the point E in the straight line EF, make (23. 1.) the angle FEH equal to the angle GEF, and join FH: Then, because GE is equal to EH, and EF common to the two triangles GEF, HEF; the two sides GE, EF are equal to the two HE, EF; and the angle GEF is equal to the angle HEF; therefore the base FG is equal (4. 1.) to the base FH: but besides FH, no straight line can be drawn from F to the circumference equal to FG: for, if there can, let it be FK; and because FK is equal to FG, and FG to FH, FK is equal to FH; that is, a line nearer to that which passes through the centre, is equal to one more remote, which is impossible. PROP. VIII. THEOR. If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; of those which fall upon the concave circumference, the greatest is that which passes through the centre; and of the rest that which is nearer to that through the centre is always greater than the more remote; But of those which fall upon the convex circumference, the least is that between the point without the circle, and the diameter; and of the rest, that which is nearer to the least is always less than the more remote: And only two equal straight lines can be drawn from the point unto the circumference, one upon each side of the least. Let ABC be a circle, and D any point without it, from which let the straight lines DA, DE, DF, DC be drawn to the circumference, whereof DA passes through the centre. Of those which fall upon the concave part of the circumference AEFC, the greatest is AD, which passes through the centre; and the line nearer to AD is always greater than the more remote, viz. DE than DF, and DF than DC; but of those which fall upon the convex circumference HLKG, the least is DG, between the point D and the diameter AG; and the nearer to it is always less than the more remote, viz. DK than DL, and DL than DH. Take (1. 3.) M the centre of the circle ABC, and join ME, MF, MC, MK, ML, MH: And because AM is equal to ME, if MD be added to each, AD is equal to EM and MD; but EM and MD are greater (20. 1.) than ED: therefore also AD is greater than ED. Again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD D are equal to FM, MD; but the angle EMD is greater than the angle FMD; therefore the base ED is greater (24. 1.) than the base FD. In like manner it may be shewn that FD is greater than CD. Therefore DA is the greatest; and DE greater than DF, and DF than DC. And because MK, KD are greater (20. 1.) than MD, and MK is equal to MG, the ' remainder KD is greater (5. Ax.) than the remainder GD, that is, GD is less than KD: And because MK, DK are drawn to the point K within the triangle MLD from M, D, the extremities of its side MD; MK, KD are less (21.1.) than ML, LD, whereof MK is equal to ML; therefore the remainder DK is less than the remainder DL: In like manner, it may be shewn that DL is less than DH: Therefore DG is the least, and DK less than DL, and DL than DH. KGB N H C M F E A Also there can be drawn only two equal straight lines from the point D to the circumference, one upon each side of the least; at the point M, in the straight line MD, make the angle DMB equal to the angle DMK, and join DB; and because in the triangles KMD, BMD, the side KM is equal to the side BM, and MD common to both, and also the angle KMD equal to the angle BMD, the base DK is equal (4. 1.) to the base DB. But, besides DB, no straight line can be drawn from D to the circumference, equal to DK; for, if there can, let it be DN; then, because DN is equal to DK, and DK equal to DB, DB is equal to DN; that is, the line nearer to DG, the least, equal to the more remote, which has been shewn to be impossible. PROP. IX. THEOR. If a point be taken within a circle, from which there fall more than two equal straight lines upon the circumference, that point is the centre of the circle. Let the point D be taken within the circle ABC, from which there fall on the circumference more than two equal straight lines, viz. DA, DB, DC, the point D is the centre of the circle. For, if not, let E be the centre, join DE, and produce it to the circumference in F, G; then FG is a diameter of the circle ABC: And because in FG, the diameter of the circle ABC, there is taken the point D which is not the centre, DG is the greatest line from it to the circumference, and DC greater (7. 3.) than DB, and DB than F DA; but they are likewise equal, which is impossible: Therefore E is not the centre of the circle ABC: In like manner it may be demonstrated, that no other point but D is the centre D therefore is the centre. DE PROP. X. THEOR. One circle cannot cut another in more than two points. B A D H K If it be possible, let the circumference FAB cut the circumference DEF in more than two points, viz. in B, G, F; take the centre K of the circle ABC, and join KB, KG, KF; and because within the circle DEF there is taken the point K, from which more than two equal straight lines, viz. KB, KG, KF, fall on the circumference DEF, the point K is (9. 3.) the centre of the circle DEF; but K is also the centre of the circle ABC; therefore the same point is the centre of two circles that cut one another, which is impossible (5. 3.). Therefore one circumference of a circle cannot cut another in more than two points. E PROP. XI. THEOR. If two circles touch each other internally, the straight line which joins their centres being produced, will pass through the point of contact. Let the two circles ABC, ADE, touch each other internally in the point A, and let F be the centre of the circle ABC, and G the centre of the circle ADE; the straight line which joins the centres F, G, being produced, passes through the point A. H A D G F E C B For, if not, let it fall otherwise, if possible, as FGDH, and join AF, AG: And because AG, GF are greater (20. 1.) than FA, that is, than FH, for FA is equal to FH, being radii of the same circle; take away the common part FG, and the remainder AG is greater than the remainder GH. But AG is equal to GD, therefore GD is greater than GH; and it is also less, which is impossible. Therefore the straight line which joins the points F and G cannot fall otherwise than on the point A; that is, it must pass through A. COR. 1. If two circles touch each other internally, the distance between their centre must be equal to the difference of their radií: for the circumferences pass through the same point in the line joining the centres. COR. 2. And, conversely, if the distance between the centres be equal to the difference of the radii, the two circles will touch each other internally. |