point D and the diameter AG; and the nearer to it is always Book III. less than the more remote, viz. DK than DL, and DL than DH. E D GB c 24. 1. Take a M the centre of the circle ABC, and join ME, MF, a 1. 3. MC, MK, ML, MH: and because AM is equal to ME, add MD to each, therefore AD is equal to EM, MD; but EM, MD are greater than ED; therefore also AD is greater than ED: b 20. 1. again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD are equal to FM, MD; but the angle EMD is greater than the angle FMD; therefore the base ED is greater than the base FD: in like manner it may be shown 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 than MD, and MK is equal to MG, the remainder KD is d than the remainder greater 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 than ML, LD, whereof MK C H E A Σ N d 4. Ax. e 21. 1. is equal to ML; therefore the remainder DK is less than the remainder DL: in like manner it may be shown, that DL is less than DH: therefore DG is the least, and DK less than DL, and DL than DH: 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 MK is equal to MB, and MD common to the triangles KMD, BMD, the two sides KM, MD are equal to the two BM, MD; and the angle KMD is equal to the angle BMD; therefore the base DK is equal to the base DB: but, besides DB, f 4. 1. there can be no straight line drawn from D to the circumference equal to DK: for, if there can, let it be DN; and because DK is equal to DN, and also to DB; therefore DB is equal to DN, that is, the nearer to the least equal to the more remote, which is impossible. If, therefore, any point, &c. Q. E. D. f Book III. a 7. 3. a 9.3. PROP. IX. PROB. IF a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle. Let the point D be taken within the circle ABC, from which to the circumference there fall 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 shall be the greatest line from it to the circumference, and DC greater a than DB, and DB than DA; but they are likewise equal, which is impossible therefore E is not the F DE G A B 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. Wherefore, if a point be taken, &c. Q. E. D. the centre of the circle DEF: but K is also the centre of the Book III. circle ABC; therefore the same point is the centre of two circles that cut one another, which is impossible b. Therefore one b 5. 3. circumference of a circle cannot cut another in more than two points. Q. E. D. PROP. XI. THEOR. IF two circles touch each other internally, the straight line which joins their centres being produced shall pass through the point of contact. A 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. For, if not, let it fall otherwise, if possible, as FGDH, and join AF, AG: and because AG, GF are greater than FA, that is, than FH, for FA is equal to FH, both being from the same centre; take away the common part FG; therefore the remainder AG is greater than the remainder GH: but AG is equal H D GE E B a 20.1. C to GD; therefore GD is greater than GH, the less than the greater, which is impossible. Therefore the straight line which joins the points F, & cannot fall otherwise than upon the point A, that is, it must pass through it. Therefore, if two circles, &c. Q. E. D. PROP. XII. THEOR. IF two circles touch each other externally, the straight line which joins their centres shall pass through the point of contact. Let the two circles ABC, ADE touch each other externally in the point A ; and let F be the centre of the circle ABC, and G the centre of ADE: the straight line which joins the points F, G shall pass through the point of contact A. For, if not, let it pass otherwise, if possible, as FCDG, and Book III. join FA, AG: and because F is the centre of the circle ABC, AF is equal to FC: also, a 20. 1. AG: but it is also less a; E therefore the straight line which joins the points F, G shall not pass otherwise than through the point of contact A, that is, it must pass through it. Therefore, if two circles, &c. Q. E. D. See N. PROP. XIII. THEOR. ONE circle cannot touch another in more points than one, whether it touches it on the inside or outside. For, if it be possible, let the circle EBF touch the circle ABC in more points than one, and first on the inside, in the points a 10.11.1. B, D; join BD, and draw a GH bisecting BD at right angles: Therefore, because the points B, D are in the circumference of b 2.3. each of the circles, the straight line BD falls within each of cCor. 1.3. them: and their centres are in the straight line GH which bi sects BD at right angles; therefore GH passes through the point d 11. 3. of contactd; but it does not pass through it, because the points B, D are without the straight line GH, which is absurd: therefore one circle cannot touch another on the inside in more points than one. K b 2.3. Nor can two circles touch one another on the outside in Book III. more than one point: for, if it be possible, let the circle ACK touch the circle ABC in the points A, C, and join AC: therefore, because the two points A, C are in the circumference of the circle ACK, the straight line AC which joins them shall fall within the circle ACK: and the circle ACK is without the circle ABC; and therefore the straight line AC is without this last circle; but, because the points A, C are in the circumference of the circle ABC, the straight line AC must be withinb the same circle, which is absurd: therefore one circle cannot touch another on the outside in more than one point: and it has been shown, that they cannot touch on the inside in more points than one. Therefore, one circle, &c. Q. E. D. A B C PROP. XIV. THEOR. EQUAL straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre, are equal to one another. Let the straight lines AB, CD, in the circle ABDC, be equal to one another; they are equally distant from the centre. F A C a 3. 3. G E D Take E the centre of the circle ABDC, and from it draw EF, EG perpendiculars to AB, CD: then, because the straight line EF, passing through the centre, cuts the straight line AB, which does not pass through the centre, åt right angles, it also bisects it: wherefore AF is equal to FB, and AB double of AF. For the same reason, CD is double of CG: and AB is equal to CD; therefore, AF is equal to CG: and because AE is equal to EC, the square of AE is equal to the square of EC; but the squares of AF, FE are equal to the square of AE, because the angle AFE is a right angle; and, for the like reason, the squares of EG, GC are equal to the square of EC: therefore the squares of AF, FE are equal to the squares of CG, GE, of which the square of AF is equal to the square of B b 47.1. |