PROP. VII. THEOR. Book III. F any point be taken in the diameter of a circle, which is not the center, of all the ftraight lines which can be drawn from it to the circumference, the greatest is that in which the center is, and the other part of that diameter is the leaft; and, of any others, that which is nearer to the line which paffes through the center is always greater than one more remote: And from the fame point there can be drawn only two ftraight 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 center: Let the center be E; of all the ftraight 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 leaft; and of the others, FB is greater than FC, and FC than FG. Join BE, CE, GE; and because two fides of a triangle are greater than the third, BE, EF are greater than BF; but AE a 20. z. is equal to EB; therefore AE, EF, BA E b 24. H D that is AF, is greater than BF: A- Alfo there can be drawn only two equal ftraight lines from the point F to the circumference, one upon each fide of the shortest C 23. I. d 4. I. Book II. fhorteft line FD: At the point E in the straight line EF, make the angle FEH equal to the angle GEF, and join FH: Then becaufe GE is equal to EH, and EF common to the two triangles GEF, HEF; the two fides GE, EF are equal to the two HE, EF; and the angle GEF is equal to the angle HEF; therefore the bafe FG is equal to the base FH: But, befides FH, no other ftraight 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 paffes through the center, is equal to one which is more remote; which is impoffible. Therefore, if any point be taken, &c. Q. E. D. PROP. VIÙ. THEOR. F any point be taken without a circle, and ftraight lines be drawn from it to the circumference, whereof one paffes through the center; of thofe which fall upon the concave circumference, the greatest is that which paffes through the center; and of the reft, that which is nearer to that through the center is always greater than the more remote: But of thofe which fall upon the convex circumference, the leaft is that between the point without the circle, and the diameter; and of the reft, that which is nearer to the leaft is always lefs than the more remote: And only two equal ftraight lines can be drawn from the point unto the circumference, one upon each fide of the leaft. Let ABC be a circle, and D any point without it, from which let the ftraight lines DA, DE, DF, DC be drawn to the circumference, whereof DA paffes through the center. Of thofe which fall upon the concave part of the circumference AEFC, the greateft is AD which paffes through the center; and the nearer to it 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 leaft is DG between the point D and the diameter AG; and the nearer to it is always Book III. lefs than the more remote, viz. DK than DL, and DL than DH. 3 H D K G& N b 20. I. C 24. I. M d 4. Ax. F E Α Take M the center of the circle ABC, and join ME, MF, a I. 3. MC, MK, ML, MH: And becaufe AM is equal to ME, add MD to each, therefore AD is equal to EM, MD; but EM, MD are greater than ED; therefore alfo AD is greater than ED : Again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD Care equal to FM, MD; but the angle EMD is greater than the angle FMD; therefore the bafe ED is greater than the base FI: In like manner it may be fhewn that FD is greater than CD: Therefore DA is the greatest; and DE greater than DF, and DF than DC: And becaufe MK, KD are greater than MD, and MK is equal to MG, the remainder KD is greater than the remainder GD, C 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 fide MD; MK, KD are lefs than ML, LD, whereof MK is equal to ML; therefore the remainder DK is lefs than the remainder DL: In like manner it may be fhewn, that DL is less than DH: Therefore DG is the leaft, and DK lefs than DL, and DL than DH: Alfo there can be drawn only two equal straight lines from the point D to the circumference, one upon each fide of the leaft: At the point M, in the straight line MD, make the angle DMB equal to the angle DMK, and join DB: And becaufe MK is equal to MB, and MD common to the triangles KMD, BMD, the two fides KM, MD are equal to the two BM, MD; and the angle KMD is equal to the angle BMD; therefore the bafe DK is equal to the bafe DB: But, befides DB, f 4. I. there can be no ftraight 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 impoffible. If therefore, any point, &c. Q.E. D. e PROP. € 21. I. Book III. PROP. IX. THEOR. I fall lines to circum F a point be taken within a circle, from which there a 7.30 ference, that point is the center of the circle. Let the point D be taken within the circle ABC, from which to the circumference there fall more than two equal ftraight lines, viz. DA, DB, DC, the point D is the center of the circle. DE For, if not, let E be the center, 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 F point D which is not the center, DG fhall be the greatest line from it to the circumference, and DC greater a than DB, and DB than DĂ: But they are likewife equal, which is impoffible: Therefore E is not the center of the circle ABC: In like manner, it may be demonftrated, that no other point but D is the center; D therefore is the center. Wherefore, if a point be taken, &c. Q. E. D. G A B NE circumference of a circle cannot cut another in more than two points. the center of the circle DEF: But K is also the center of the Book III. circle ABC; therefore the fame point is the center of two circles that cut one another, which is impoffible. Therefore one b 5. 3.! circumference of a circle cannot cut another in more than two points. Q. E. D. IF two circles touch each other internally, the straight line which joins their centers being produced fhall 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 center of the circle ABC, and G the center of the circle ADE: The ftraight line which joins the centers F, G, being produced, paffes through the point A. For, if not, let it fall otherwife, if poffible, as FGDH, and join AF, AG: And becaufe AG, GF are greater than FA, that is, than FH, for FA is equal to FH, both being from the fame center; take away the common part FG; therefore the remain HA GF E B der 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 impoffible. Therefore the straight line which joins the points F, G cannot fall otherwife 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 centers fhall 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 center of the circle ABC, and G the center of ADE: The straight line which joins the points F, G fhall pass through the point of contact A. For, if not, let it pafs otherwife, if poffible, as FCDG, and a 20. s. |