If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center 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.* Draw *CA* from *C* at right angles to *DE.*

I say that the center of the circle is on *AC.*

For suppose it is not, but, if possible, let *F* be the center, and join *CF.*

Since a straight line *DE* touches the circle *ABC,* and *FC* has been joined from the center to the point of contact, *FC* is perpendicular to *DE.* Therefore the angle *FCE* is right.

But the angle *ACE* is also right, therefore the angle *FCE* equals the angle *ACE,* the less equals the greater, which is impossible.

Therefore *F* is not the center of the circle *ABC.*

Similarly we can prove that neither is any other point except a point on *AC.*

Therefore *if a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn.
*

Q.E.D.