Commentary on Proposition III.6¶
Back to Proposition III.6.
The English editions enunciate this proposition of circles touching internally, but the word (ἐντός) is a mere interpolation, which was no doubt made because Euclid’s figure showed only the case of internal contact. The fact is that, in his usual manner, he chose for demonstration the more difficult case, and left the other case (that of external contact) to the intelligence of the reader. It is indeed sufficiently self-evident that circles touching externally cannot have the same centre; but Euclid’s proof can really be used for this case too.
Camerer remarks that the proof of III. 6 seems to assume tacitly that the points $E$ and B cannot coincide, or that circles which touch internally at C cannot meet in any other point, whereas this fact is not proved by Euclid till III. 13. But no such general assumption is necessary here; it is only necessary that one line drawn from the assumed common centre should meet the circles in different points; and the very notion of internal contact requires that, before one circle meets the other on its inner side, it must have passed through points within the latter circle.