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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.
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.