Prove that a line that intersects a circle at only one point is a tangent line.

This is the converse of one that I've already proved. The previous proof was that A line is a tangent line to a circle IFF it intersects a circle at only one point.

The goal here is to prove that in a circle O where there is a line L through a point A on the circle that OA is perpendicular to L.

Question

Prove that a line that intersects a circle at only one point is a tangent line.

This is the converse of one that I've already proved. The previous proof was that A line is a tangent line to a circle IFF it intersects a circle at only one point.

The goal here is to prove that in a circle O where there is a line L through a point A on the circle that OA is perpendicular to L.

cgreenwade2000 · Answer

I attempt to solve by contradiction but I don't think I can prove anything by assuming that AO is not perpendicular to L.

anonymous · Answer

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cgreenwade2000 · Answer

That is correct but from there, I'd just be assuming things. I tried using P' as the intersection of the circle and OP.

anonymous · Answer

I think it is a proposition that is assumed to be true in order to test the validity of another propositions in the geometry. I haven't yet seen any proof to this theorem.

cgreenwade2000 · Answer

Neither have I. 

But I think I'm going to continue in this direction. After creating P' I see that triangle OAP' is isosceles. Maybe I can work with this.

anonymous · Answer

I don't think it works. you don't need to proof it. it is a lemma...

cgreenwade2000 · Answer

-_- Ok

cgreenwade2000 · Answer

Thanks.

anonymous · Answer

Anyway...

cgreenwade2000 · Answer

It's not really something that I need to show. It's more that it was bothering me not being able to.