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Proposition 6

If two circles touch one another, then they do not have the same center.
Let the two circles ABC and CDE touch one another at the point C. I say that they do not have the same center. For, if possible, let it be F. Join FC, and draw FEB through at random. |
Then, since the point F is the center of the circle ABC, FC equals FB. Again, since the point F is the center of the circle CDE, FC equals FE. | I.Def.15
But FC was proved equal to FB, therefore FE also equals FB, the less equals the greater, which is impossible. Therefore F is not the center of the circles ABC and CDE.
Therefore if two circles touch one another, then they do not have the same center.
Q.E.D.

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As mentioned before, this proposition is almost the same as the previous. Both could be included in one statement: circles that meet don't have the same center, or the contrapositive: concentric circles don't meet.

This propostion is not used in the rest of the Elements.

Book III Introduction - Proposition III.5 - Proposition III.7.

© 1996
D.E.Joyce
Clark University