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.
Therefore F is not the centre of the circle ABC.
Similarly we can prove that neither is any other point except a point on AC.