Euclid's Elements, Book III, Definitions 2 and 3

Definitions 2 and 3

Def. 2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

Def. 3. Circles are said to touch one another which meet one another but do not cut one another.

Consider circle BCD in the figure. The line EF touches this circle at the point C. Another expression for the same thing is that EF is tangent to the circle at C.

Two circles can touch each other either internally or externally. Circle HKL touches circle BCD externally, while circle DMN touches circle BCD internally.

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Book III introduction