◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--3-circles / Proof: By Euclid

Proof: By Euclid

(related to Proposition: 3.06: Touching Circles have Different Centers)

• For, if possible, let $F$ be (the common center), and let $FC$ have been joined, and let $FEB$ have been drawn through (the two circles), at random.
• Therefore, since point $F$ is the center of the circle $ABC$, $FC$ is equal to $FB$.
• Again, since point $F$ is the center of the circle $CDE$, $FC$ is equal to $FE$.
• But $FC$ was shown (to be) equal to $FB$.
• Thus, $FE$ is also equal to $FB$, the lesser to the greater.
• The very thing is impossible.
• Thus, point $F$ is not the (common) center of the circles $ABC$ and $CDE$.
• Thus, if two circles touch one another then they will not have the same center.
• (Which is) the very thing it was required to show.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"