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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.
In the O draw the str. line AB, and bisect it
bisect CE in F:
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. Zs.
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. Ls
to AB from its extremity A, shall fall without
Fig. 1. If not, let it fall if possible within as AC.
: _DAC = _ ACD, Prop. 5.
but _DAC=rt. L, Hyp.
.. ZACD=rt. L, Prop. 16. :: _DAC+ LACD =12 rt. Zs, which is
.. if AC | AB, AC does not fall within the O. In the same manner it may be shown, that AC does not fall upon the O", .. falls
without the O as AE, fig. 2. Fig. 2. 2ndly. Between AE and the O“, no str.
line can be drawn from A, which does not cut the O.
For if possible, let AF be between them.
Draw DG I AF meeting the oce in H; Const. :: LAGD = rt. L, Prop. 16. and LDAG < rt. L,
DA > DG,
but DA = DH,
.. no str. line can be drawn from A between AE and the oce which does not cut the
CoR.-A str. line drawn perpendicular to the diam. of a O from the Oce at the extremity of the diam. ; touches the o in one pt. only, and being produced does not cut it, and is therefore a tangent to the circle at that Del. 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 O BCD; it is required to draw a str. line from A, which shall touch the O.