Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/101

Rh 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.

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.