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 O ; it is required to
find its cr.

In the O draw the str. line AB, and bisect it

in D;
draw DC I AB,
prod. CD to E in O,

bisect CE in F:
Then F is the Cr. of O ACB.

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
the O.
Fig i.

Fig 2.

A

15 def.

Fig. 1. 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, 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,

Prop. 18.

15 def.

but DA = DH,
.. DH > 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

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.

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 O BCD; it is required to draw a str. line from A, which shall touch the O.