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Euclid: Elementa
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PROPOSITION 19. 
THEOR. 17. PROPOS. 19. 
第十九題 
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. 
SI circulum tetigerit recta quæpiam linea, a contactu autem recta linea ad angulos rectos ipsi tangenti excitetur: In excitata erit centrum circuli. 
直線切圜。圜內作切線之垂線。則圜心必在垂線之內。 
For let a straight line DE touch the circle ABC at the point C, and from C let CA 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 be joined. 
 
 
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. 
             
             
Therefore etc.  Q. E. D. 
   
   
 
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