Suppose I have a line in the plane, and two points not on the line. How can I prove that there is a unique generalized circle (i.e. circle or line) passing through the two points which intersects the line at right angles? (We can suppose that the two points are on the same side of the line if necessary.)
Methods of complex analysis/conformal geometry are preferable. Bonus points if you can explain when there is a circle through two given points making an angle $\theta$ with the line, and how many there are.
Background: In the Poincare half-space model of $H^2$, the shortest distance between two points is an arc of such a circle, and I can't seem to prove that there should be only one (and at least one) such arc.
let us call the two given points $A, B$ and the line $l.$ if $AB$ is orthogonal to $l,$ then the line $AB$ itself is the generalized circle that is orthogonal to $l.$
so we assume that $AB$ is not orthogonal to $l.$ draw the orthogonal bisector of $AB$ and let it cut $l$ at $O.$ the the circle centered at $O$ and radius $OA = OB$ is the unique circle through the points $A, B$ and orthogonal to the line $l.$