If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.

For let a straight line *DE* touch the circle *ABC* at the point *C.* Take the center *F* of the circle *ABC,* and join *FC* from *F* to *C.*

I say that *FC* is perpendicular to *DE.*

For, if not, draw *FG* from *F* perpendicular to *DE.*

Then, since the angle *FGC* is right, the angle *FCG* is acute, and the side opposite the greater angle is greater, therefore *FC* is greater than *FG.*

But *FC* equals *FB,* therefore *FB* is also greater than *FG,* the less greater than the greater, which is impossible.

Therefore *FG* is not perpendicular to *DE.*

Similarly we can prove that neither is any other straight line except *FC.* Therefore *FC* is perpendicular to *DE.*

Therefore *if a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.*

Q.E.D.