† 9 Ax. 1. 24. 1. EF, that is AF, is greater than BF: again, because BE is equal to CE, and FE common to the triangles BEF, CEF, the two sides BE, EF are equal to the two CE, EF, each to each; but the angle BEF is greater than the angle CEF; therefore the base BF is greater *than the base CF: for the same reason, CF is greater than GF: again, because GF, FE are greater * than EG, and EG is equal to ED, therefore GF, FE are greater than ED: take away the common part FE, and the remainder GF is greater than the remainder FD: † 5 Ax. 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. * 20. 1. 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 the angle FEH equal to the angle FEG, and join FH: *23. 1. then, because GE is equal to EH, and EF common to the + 15 Def. 1. two triangles GEF, HEF, the two sides GE, EF are equal to * to the 4. 1. the two HE, EF, each to each; and the angle GEF is equal † † Constr. to the angle HEF; therefore the base FG is equal base FH but, besides FH, no other 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 † 1 Ax. passes through the centre, is equal to one which is more remote; which has been proved to be impossible. Therefore, if any point be taken, &c. Q. E. D. PROPOSITION 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 the one passing through the centre is always greater than one more remole; 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 one more remote; and only two equal straight lines can be drawn from the same 1. 3. +2 Ax. * 20. 1. point to the circumference, one upon each side of the least line. 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 shall be DA, which passes through the centre; and the nearer to it shall be greater than the more remote, viz. DE greater than DF, and DF greater than DC: but of those which fall upon the convex circumference HLKG, the least shall be DG, between the point D and the diameter AG; and the nearer to it shall be less than the more remote, viz. DK less than DL, and DL less than DH. Take M the centre of the circle ABC, and join ME, MF, 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. Again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD are equal to FM, MD, each to each: 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 shewn, that FD is greater than CD: therefore DA is the greatest, and DE greater than DF, and DF greater than DC. And because MK, KD are greater* † 15 Def. 1. than MD, and MK is equal to MG, the remainder KD is greater than the remainder † 9 Ax. • 24. 1. • 20. 1. ⚫ 5 Ax. * 21. 1. ↑ 15 Def. 1. ↑ 5 Ax. † 23. 1. GD; that is, GD is less than KD: and because MLD is a triangle, and from the points M, D, the extremities of its side MD, the straight lines MK, DK are drawn to the point K within the triangle, therefore MK, KD are less than ML, LD: but 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 less 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 line. 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, each to each; and the angle KMD is equal to the angle BMD; † Constr. therefore the base DK is equal to the base DB: but, besides • 4. 1. DB, there can be no straight line drawn from D to the circumference equal to DK: for if there can, let it be DN: and be cause DK is equal to DN, and also to DB, therefore DB is equal to DN; that is, a line nearer to the least, is equal to one more remote ; which has been proved to be impossible. therefore, any point, &c. Q. E. D. PROPOSITION IX. If, THEOR.-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 shall be the centre of the circle. DE F 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 17 Def. 1. 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 is greater than DB, and DB greater than 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. point be taken, &c. Q. E. D. PROPOSITION X. • 7.3. B + Hyp. Wherefore, if a THEOR.-One circumference of a circle cannot cut another in more than two points. 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. † 3. 3. Then because K is the centre of the circle ABC, therefore KB, † 15 Def. 1. KG, KF are all equal † to each other: and because within the *9.3. * 5. 3. circle DEF, there is taken the point K, from which to the * K is the centre of the circle DEF but K is B D H K F also the centre † of the circle ABC; there- E Q. E. D. † 20.1. † 5 Ax. † 15 Def. 1. PROPOSITION 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. 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, shall pass through the point A. For, if not, let it fall otherwise, if possible, as FGDH, and join AF, AG. Then, because two sides of a triangle are together greater + than the third side, therefore FG, GA are H EDGE E B † 15 Def. 1. greater than FA: but FA is equal to FH; therefore FG, GA are greater than FII; take away the common part FG; therefore the remainder AG is greater than the remainder GH: but AG is equal † 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, G, being produced, cannot fall otherwise than upon the point A, that is, it must pass through it. Therefore, if two circles, &c. Q. E. D. PROPOSITION 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 join FA, AG. And because F is the centre of the circle ABC, FA is equal to FC: also, because G is the centre of the circle ADE, GA is equal to GD: therefore FA, AG are equal to FC, DG; wherefore the whole FG is greater than FA, AG: but it is also less; which is impossible: therefore the straight line which joins the points F, G, cannot pass otherwise than through the point of contact A, that is, it must pass through it. Therefore, if two circles, &c. Q. E. D. B A † 2 Ax. * 20. 1. PROPOSITION XIII. THEOR.-One circle cannot touch another in more points than See N. one, whether it touches it on the inside or outside. G H B H * 10. 11. 1. A D 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 B, D: join BD, and draw GH bisecting BD at right angles: therefore, because the points B, D are in the circumferences of each of the circles, the straight line BD falls within each * of them; therefore their centres are * in the straight line GH Cor. 1. 3. which bisects BD at right angles; therefore GH passes through the point of contact: 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, * Nor can two circles touch one another on the outside in more than one point. For if it be possible, let the circle ACK touch the circle ABC in the points A, C; join AC: therefore because the two points A, C are in the circumference of the circle ACK, the straight line AC which joins them, * 2.3. * 11. 3. |