Proof: By Euclid
(related to Proposition: 3.18: Radius at Right Angle to Tangent)
- For if not, let $FG$ have been drawn from $F$, perpendicular to $DE$ [Prop. 1.12].
- Therefore, since angle $FGC$ is a right angle, (angle) $FCG$ is thus acute [Prop. 1.17].
- And the greater angle is subtended by the greater side [Prop. 1.19].
- Thus, $FC$ (is) greater than $FG$.
- And $FC$ (is) equal to $FB$.
- Thus, $FB$ (is) also greater than $FG$, the lesser than the greater.
- The very thing is impossible.
- Thus, $FG$ is not perpendicular to $DE$.
- So, similarly, we can show that neither (is) any other (straight line) except $FC$.
- Thus, $FC$ is perpendicular to $DE$.
- Thus, if some straight line touches a circle, and some (other) straight line is joined from the center (of the circle) to the point of contact, then the (straight line) so joined will be perpendicular to the tangent.
- (Which is) the very thing it was required to show.
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Footnotes
