G.104 touching circles have distinct centers — Heath

III.6

If two circles touch one another, they will not have the same centre.

For let the two circles ( A B C ), ( C D E ) touch one another at the point C;

I say that they will not have the same centre.

For, if possible, let it be F; let / F C / be joined, and let / F E B / be drawn through at random.

Then, since the point F is the centre of the circle ( A B C ),

/ F C / is equal to / F B /.

Again, since the point F is the centre of the circle ( C D E ),

/ F C / is equal to / F E /.

But / F C / was proved equal to / F B /; therefore / F E / is also equal to / F B /, the less to the greater: which is impossible.

Therefore F is not the centre of the circles ( A B C ), ( C D E ).

Therefore etc. Q. E. D.

Dependency Graph