I have seen proofs for proving if a line is tangent to the circle, then it has to be perpendicular to the radius of the circle. But I want to know how you can prove the other direction of this statement since this is biconditional statement. Start with the hypothesis that some line is perpendicular to the radius of the circle, then prove it must be a tangent line to the circle.
If I were to use the proof by contradiction, do I have to consider the case where the line intersects the circle at two points as well as the case when the line is not meeting the circle at all?
Best Answer
The hypothesis should be that line $t$ is perpendicular to the radius $OT$ at point $T$ on the circle of center $O$. In that case, if $P$ is another point on $t$, different from $T$, then we have $OP>OT$ (hypotenuse is greater then leg). It follows that all points of $t$, except $T$, are external to the circle, and $t$ is thus a tangent.