If two circles touch one another internally
, and their centres be taken
, the straight line joining their centres
, if it be also produced
, will fall on the point of contact of the circles
For let the two circles ABC
touch one another internally at the point A
, and let the centre F
of the circle ABC
, and the centre G
, be taken; I say that the straight line joined from G
and produced will fall on A
For suppose it does not, but, if possible, let it fall as FGH
, and let AF
Then, since AG
are greater than FA
, that is, than FH
be subtracted from each; therefore the remainder AG
is greater than the remainder GH
is equal to GD
; therefore GD is also greater than GH, the less than the greater: which is impossible.
Therefore the straight line joined from F
will not fall outside; therefore it will fall at A on the point of contact.
Therefore etc. Q. E. D.