If a straight line touch a circle
, and from the point of contact a straight line be drawn at right angles to the tangent
, the centre of the circle will be on the straight line so drawn
For let a straight line DE
touch the circle ABC
at the point C
, and from C
be drawn at right angles to DE
; I say that the centre of the circle is on AC
For suppose it is not, but, if possible, let F
be the centre, and let CF
Since a straight line DE
touches the circle ABC
, and FC
has been joined from the centre to the point of contact, FC is perpendicular to DE; [III. 18] therefore the angle FCE is right.
But the angle ACE
is also right; therefore the angle FCE is equal to the angle ACE, the less to the greater: which is impossible.
is not the centre of the circle ABC
Similarly we can prove that neither is any other point except a point on AC
Therefore etc. Q. E. D.