Theorem. If a line (CD) touch a circle, the line (OC) from the centre to the point of contact is perpendicular to it.
Dem.—If not, suppose another line OG drawn from the centre to be perpendicular to CD. Let OG cut the circle in F. Then because the angle OGC is right (hyp.) the angle OCG [I. xvii.] must be acute. Therefore [I. xix.] OC is greater than OG; but OC is equal to OF [I. Def. xxxii.]; therefore OF is greater than OG—that is, a part greater than the whole, which is impossible. Hence OC must be perpendicular to CD.
Or thus: Since the perpendicular must be the shortest line from O to CD, and OC is evidently the shortest line; therefore OC must be perpendicular to CD.