previous next

PROPOSITION 18.

If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.

For let a straight line DE touch the circle ABC at the point C, let the centre F of the circle ABC be taken, and let FC be joined from F to C; I say that FC is perpendicular to DE.

For, if not, let FG be drawn from F perpendicular to DE.

Then, since the angle FGC is right,

the angle FCG is acute; [I. 17]
and the greater angle is subtended by the greater side; [I. 19]
therefore FC is greater than FG.

But FC is equal to FB;

therefore FB is also greater than FG, the less than the greater: which is impossible.

Therefore FG is not perpendicular to DE.

Similarly we can prove that neither is any other straight line except FC;

therefore FC is perpendicular to DE.

Therefore etc. Q. E. D.1

1 1

2 the tangent, ἡ ἐφαπτομένη.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

The National Science Foundation provided support for entering this text.

Purchase a copy of this text (not necessarily the same edition) from Amazon.com

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 United States License.

An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.

load focus Greek (J. L. Heiberg, 1883)
hide Display Preferences
Greek Display:
Arabic Display:
View by Default:
Browse Bar: