e 21. I. £ 4. I. angle MLD from M, D, the extremities of its fide MD; MK, KD are lefse 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 lefs than DH: Therefore DG is the leaft, and DK less than DL, and DL than DH. Alfo there can be drawn only two equal ftraight lines from the point D to the circumference, one upon each fide of the leaft at the point M, in the ftraight line MD, make the angle DMB equal to the angle DMK, and join DB; and because in the triangles KMD, BMD, the fide KM is equal to the fide BM, and MD common to both, and also the angle KMD equal to the angle BMD, the bafe DK is equal f to the bafe DB. But, befides DB, no ftraight line can be drawn from D to the circumference, equal to DK: for, if there can, let it be DN; then, becaufe DN is equal to DK, and DK equal to DB, DB is equal to DN; that is, the line nearer to DG, the leaft, equal to the more remote, which has been fhewn to be impoffible. If, therefore any point, &c. Q. E. D. PROP. Book III. IF PROP. IX. THEOR. F a point be taken within a circle, from which there fall more than two equal ftraight 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 ftraight 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 F of the circle ABC, there is taken the point D which is not the centre, DG fhall be the greatest line from it to the circumference, and DC greater a than DB, and DE B DB than DA; but they are likewife equal, which is impoffible: 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. Wherefore, if a point be taken, &c. Q. E. D. a 7.3. NE circle cannot cut another in more than two If Book III. 2 9. 3. b 5.3. If it be poffible, let the cir- B cumference DEF, the point K is the centre of the circle a 20. I. F two circles touch each other internally, the ftraight line which joins their centres being produced, will pafs 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 ftraight line which joins the centres F, G, being produced, paffes through the point A. a For, if not, let it fall otherwife, if poffible, as FGDH, and join AF, AG: And because AG, GF are greater than FA, that is, than FH, for FA is equal to FH, being radii of the fame circle; take away the common part FG, and the remain H D F E B der AG is greater than the remainder GH. But AG is equal to GD, therefore GD is greater than GH; and it is alfo lefs, which is impoffible. Therefore the ftraight line which joins the points F and G cannot fall otherwise than on the point A; that is, it must pass through A. Therefore, if two circles, &c. Q. E. D. PROP. PROP. XII. THEOR. F two circles touch each other externally, the which joins their centres pals 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 ftraight line which joins the points F, G hall pass through the point of contact A. B E For, if not, let it pass otherwife, if poffible, as FCDG, and join FA, AG: and because F is the centre of the circle ABC, AF is equal to FC: Alfo because G is the centre of the circle ADE, AG is equal to GD. Therefore FA, AG are equal to FC, DG; wherefore the CAD F G whole FG is great Book III. er than FA, AG; but it is alfo lefs a, which is impoffible: a 20. 1. Therefore the ftraight line which joins the points F, G, cannot pass otherwife than through the point of contact A; that is, it paffes through A. Therefore, if two circles, &c. Q.E. D. ON PROP. XIII. THEOR. touch NE circle cannot touch another in more points than one, whether it touches it on the infide or outfide. 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; join BD, and draw a GII, bifecting BD at right a 10. 11. 1. angles : G Book III. angles: Therefore, because the points B, D are in the circum. ference of each of the circles, the ftraight line BD falls within each of them; and their centres are in the ftraight line c Cor. 1. 3. GH which bifects BD at right angles: Therefore GH paffes d 11.3. through the point of contact d; but it does not pass through it, because the points B, D are without the ftraight line GH, which is abfurd: Therefore one circle cannot touch another in the infide in more points than one. K Nor can two circles touch one another on the outfide in more than one point: For, if it be poffible, 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 ftraight line AC which joins them fhall fall within. the circle ACK: And the circle ACK is without the circle ABC; and therefore the ftraight line AC is without this laft circle; but, because the points A, C are in the circumference of the circle ABC, the straight line AC must be within the fame circle, which is abfurd: Therefore a circle cannot touch another on A B the outfide in more than one point; and it has been shewn, that a circle cannot touch another on the infide in more than one point. Therefore, one circle, &c. Q. E. D. PROP. |