c Book III. FD. at the point E in the straight line EF, make the angle FEH in equal to the angle GEF, and join FH. C. 23. I. d. 4. I. then because GE is equal to EH, and A B DH 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 thro' the center is equal to one which is more remote; which is impoffible. Therefore if any point be taken, &c. Q. E. D. IF PROP. VIII. THE OR. F any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one paffes thro' the center; of those which fall upon the concave circumference the greateft is that which paffes thro' the center; and of the reft, that which is nearer to that thro' the center is always greater than the more remote. but of thofe which fall upon the convex circumference, the leaft is that be tween the point without the circle, and the diameter; and of the reft, that which is nearer to the least 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 straight lines DA, DE, DF, DC be drawn to the circumference, whereof DA paffes thro' the center. of those which fall upon the concave part of the circumference AEFC, the greatest is AD which paffes thro' 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 feaft is DG between the point D and the diameter AG; and the Book III. D 24. Take M the center of the circle ABC, and join ME, MF, MC, 1. 1. 3. 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 b. 20. 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 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 shewn that FD is greater than CD. therefore DA is the greateft; 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 greater than the remainder GD, that is, GD is lefs than KD. and because MK, DK are drawn to the point K within the F H GB N M d. 4. Azi triangle MLD from M, D the ex tremities of its fide MD; MK, KD E A are less than ML, LD, whereof c. 21. Es MK is equal to ML, therefore the remainder DK is lefs than the Book III. 2. 7. 3. 2.9.3. IF F a point be taken within a circle, from which there fall more than two equal ftraight lines to the circumference, 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 straight lines, viz. DA, DB, DC. the point D is the center of the circle. For if not, let E be the center, DE G join DE and produce it to the cir- A B circle DEF. but K is alfo the center of the circle ABC; therefore Book III. the fame point is the center of two circles that cut one another, which is impoffible. Therefore one circumference of a circle b. s. j. cannot cut another in more than two points. Q. E. D. PROP. XI. THEOR. IF Let the two circles ABC, ADE touch each other internally in For if not, let it fall otherwife, if H GF E B is greater than the remainder GH. but AG is equal to GD, there- IF PROP. XII. THE OR. 2. 20. I. two circles touch each other externally, the ftraight line which joins their centers fhall pass thro' 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 shall pass thro' the point of contact A. For if not, let it pass otherwise, if poffible, as FCDG, and join Book III. FA, AG. and because F is the center of the circle ABC, AF is Bee N. but it is alfo lefs; which is impoffible. therefore the straight line which joins the points F, G fhall not pafs otherwise than thro' the point of contact A, that is, it must pass thro' it. Therefore if two circles, &c. Q. E. D. ON PROP. XIII. THEOR. NE circle cannot touch another in more points than whether it touches it on the infide or outfide. one, For, if it be poffible, let the circle EBF touch the circle ABC in more points than one, and firft on the infide, in the points B, D; . 10. 11. 1. join BD, and draw GH bifecting BD at right angles. therefore because the points B, D are in the circumference of each of the b. 2. 3. circles, the ftraight line BD falls within beach of them. and their c. Cor. 1. 3. centers are in the straight line GH which bifects BD at right d. 1. 3. ་་་ angles; therefore GH paffes thro' the point of contact. but it does not pass thro' it, because the points B, D are without the straight line GH, which is abfurd. 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 |