DEF.-A straight line is said to touch a circle when it meets the circle, and being produced does not cut it. This line is said to be a tangent to the circle at the point of contact. PROP. XLIX. PROB, 1. 3 Eu. To find the centre of a given circle. Let ABC be a given ; it is required to find its cr. In the draw the str. line AB, and bisect it in D; draw DC AB, prod. CD to E in O, bisect CE in F: Then F is the Cr. of ACB. If not, let if possible G be the Cr. Join GA, GD, GB. In As ADG, BDG, (DA = DB, GA= GB, Const. 15 def. FDB = /GDB, = i. e. greater less, which is impossible. .. G is not the Cr. of O ACB. And no other point but F is the Cr.; which was to be found. COR.-If in a O, a str. line bisect another one at rt. s, the Cr. of is in the bisecting line. PROP. L. THEOR. 3. 3 Eu. If a straight line drawn through the centre of a circle, bisect a straight line in it which does not pass through the centre, it shall cut it at right angles: and if it cut it at right angles, it shall bisect it. Let ABC be a O, and let CD passing through the Cr. bisect AB not passing through it, in the point F; then CD shall cut AB at rt. s. 1st. And Take E the Cr. of O ABC, In As AFE, BFE, (AF = FB, EA = EB, EF common, they are adj. Zs, .. CD cuts AB at rt. Ls. 2ndly. Let CD cut AB at rt. s: then AFE = rt. / BFE, EF common, AF= i. e. CD bisects AB, in pt. F. Wherefore, if a str. line, &c. Hyp. 15 def. 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 the O. 15 def. Prop. 5. Нур. If not, let it fall if possible within as AC. Draw DC to the point C, where AC meets the Oce; DA = DC, DAC ACD, = but DAC = rt. L, rt. Zs, which is Prop. 16... ≤ DAC+/ACD = impossible. Fig. 2. Const. Prop. 16. .. if AC AB, AC does not fall within the 2ndly. Between AE and the ce .. falls c, no str. line can be drawn from A, which does not cut the O. For if possible, let AF be between them. Draw DGAF meeting the ce in H; AGD rt. L, but .. DH > DG; DA = DH, 15 def. i. e. less > greater, which is impossible. .. no str. line can be drawn from A be tween AE and the ce which does not cut the COR.-A str. line the diam. of a drawn perpendicular to from the O ce at the ex tremity 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. 67. 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. Prop. 49. AFG, Find E the Cr. of the O, join AE meeting in D. From Cr. E at dist. EA, descr. draw DF EA, also EBF and AB; then AB shall touch BCD. |