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We study certain points significant for the hyperbolic geometry of the unitdisk. We give explicit formulas for the intersection points of the Euclideanlines and the stereographic projections of great circles of the Riemann spherepassing through these points. We prove several results related to collinearityof these intersection points, offer new ways to find the hyperbolic midpoint,and represent a formula for the chordal midpoint. The proofs utilize Gr\"obnerbases from computer algebra for the solution of polynomial equations.

Collinearity of points on Poincar\'e unit disk and Riemann sphere