1st. Take E the Cr. of ○ ABC, In As AFE, BFE, EA = EB, EF common, ZAFE = ZBFE. Hyp. 15 def. And they are adj. s, .. CD cuts AB at rt. S. 2ndly. Let CD cut AB at rt. s: then shall AF FB. i. e. CD bisects AB, in pt. F. Wherefore, if a str. line, &c. Prop. 7. 10 def. 15 def. Prop. 5. Prop. 25. PROP. LI. THEOR. 16. 3 Eu. A straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle. 1st. Let ABC be a O, with Cr. D and diam. AB: then the str. line drawn at rt. s to AB from its extremity A, shall fall without Fig. 1. 15 def. If not, let it fall if possible within as AC. Draw DC to the point C, where AC meets Prop. 16... ≤ DAC+ ≤ ACD ={' Fig. 2. Const. 2 rt. Zs, which is impossible. .. if AC AB, AC does not fall within the ce .. falls ce no str. 2ndly. Between AE and the O line can be drawn from A, which does not cut the O. For if possible, let AF be between them. ce Draw DGAF meeting the " in H; AGD=rt. L, DA > DG, i. e. less > greater, which is impossible. .. no str. line can be drawn from A between AE and the Oce which does not cut the O. COR.-A str. line drawn perpendicular to the diam. of a from the Oce at the extremity of the diam. ; touches the in one pt. only, and being produced does not cut it, and is therefore a tangent to the circle at that Des. pa. extremity. PROP. LII. PROB. 17. 3 En. To draw a tangent to a given circle, from a given point, either without or in the circumference of the circle. 1st. Let A be a given point without the given BCD; it is required to draw a str. line from A, which shall touch the O. A 67. Find E the Cr. of the O, join AE meeting in D. From Cr. E at dist. EA, descr. © AFG, draw DFEA, also EBF and AB; then AB shall touch BCD. Prop. 49. 15 def. E is the Cr. of Os BCD, AFG. ..AE, EB = FE, ED, ea. to ea. Cor. Cor. Prop. 51. ZEBA = < EDF, but EDF = rt. ▲, ZEBA rt. ; and EB is drawn from the Cr. and AB EB, .. AB touches the and is drawn from A; which was to be done. 2ndly. If the given point be in the Oce of the, as the point D. Draw DE to the Cr. E, DF DE, DF touches the O. PROP. LIII. THEOR. 18. 3 Eu. If a straight line touch a circle; the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle. Let str. line DE touch O ABC in C; take the centre F, and draw str. line FC; then shall FC DE. In the like manner it may be shown, that no other line but FC is DE, Prop. 18. 15 def. PROP. LIV. THEOR. 19. 3 Eu. If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line. Let str. line DE touch O ABC in C; draw CAL DE: then the Cr. of shall be in CA. |