|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.|
This propostion is not used in the rest of the Elements.
Next proposition: III.7