points B, C: they shall not have the same centre.
For, if it be possible, let E be their centre; join EC, and draw any straight line EFG meeting the circumferences at F and G.
Then, because E is the centre of the circle ABC, EC is equal to EF. [I. Definition 15.
Again, because E is the centre of the circle CDG, EC is equal to EG. [I. Definition 15.
But EC was shewn to be equal to EF;
therefore EF is equal to EG,[Axiom 1.
the less to the greater; which is impossible.
therefore E is not the centre of the circles ABC, CDG.
Wherefore, if two circles &c. q.e.d.
PROPOSITION 6. THEOREM.
If two circles touch one another internally they shall not have the same centre.
Let the two circles ABC, CDE touch one another internally at the point G: they shall not have the same centre.
For, if it be possible, let F be their centre; join FC, and draw any straight line FEB, meeting the circumferences at E and B.
Then, because F is the centre of the circle ABC, FC is equal to FB. [I. Def. 15.
Again, because F is the centre of the circle CDE,
FC is equal to FE. [I. Definition 15.
But FC was shewn to be equal to FB
therefore FE is equal to FB,
the less to the greater; which is impossible.
Therefore F is not the centre of the circles ABC, CDE.
Wherefore, if two circles &c. q.e.d.
