Which are the stations in Circle Line?

Given a hyperbolic circle C, O its center, and A a point on C, how do I prove that a line through A perpendicular to the line OA is tangent to the circle?

• I know it is enough to show that it doesn't intersect the circle a second time. I looked at the case when drawing the perpendicular to line OA through O intersects at B and showed by proof by contradiction that the perpendicular through A cannot intersect the circle at B.  I am stuck when trying to find the other cases, any hints would be appreciated.

• Answer:

  Given that the line is perpendicular to OA through A, assume that the perpendicular intersects C at B.  Drop the perpendicular line from B to OA to intersect OA at C.  Make the triangle with vertices A, B, and C.  The angle at A is 90 degrees, as well as the angle at C.  This is contradiction because the angle sum of triangles in hyperbolic geometry is less than 180.  Basically the same thing that you did when dropping the perpendicular through O.  In that case, O = C.