Constructing a hyperbolic geodesic in the Poincare disk

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Recall that a hyperbolic line (or geodesic) through two points P and Q in the Poincare disk is an arc of the unique circle which (a) passes through P and Q, and which is orthogonal to the circle bounding the Poincare disk.
The picture below contains an example of a hyperbolic geodesic through the points P and Q. The centre of the Poincare disk is labelled O. Can you construct a hyperbolic geodesic through the two points A and B?

To help you get a feel for what the solution should be, there is already a tool for constructing hyperbolic geodesics. Try and construct them yourself, using only the other given tools.
Hint. There is a tool for constructing the inverse of a point through a circle. Can you find a way to use this?