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 I DE.

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 CA I DE: then the Cr. of shall be in CA.

LE

For if not, if possible let F be the Cent. of O,

join CF; Then :: DE touches O ABC, and FC is drawn from the Cr. to point of

contact, Prop. 53.

i. FC I DE,

.. ZFCE =rt. L, Hyp.?

but LACE=rt. L,

.: LFCE= LACE, i. e. less = greater, which is impossible, .. F is not Cr. of O ABC.

In like manner it may be shown that no other pt. which is not in CA, is the Cr. of O ABC;

i. e. the Cr. of O, is in CA. Therefore, if a str. line, &c.

PROP. LV. THEOR. 20.3 Eu.

The angle at the centre of a circle is double of

the angle at the circumference upon the same arc, that is, upon the same part of the circumference.

Let ABC be a O; BEC an 2 at the Cr. E, and BAC an , at the Oce, having the same arc BC for their base: then shall Z BEC = 2 BAC.

The angles in the same segment of a circle are

equal to one another.

Let ABCD be a O, and Zs BAD, BED in the same segment BAED: then shall Z BAD =ZBED.

PROP. LVII. THEOR. 22. 3 Ea. The opposite angles of any quadrilateral