In hyperbolic geometry, the Ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line.

Let a < b < c < d be four distinct points on the abscissa of the Cartesian plane. Let p and q be semicircles above the abscissa with diameters ab and cd respectively. Then in the upper half-plane model HP, p and q represent ultraparallel lines.

Compose the following hyperbolic motions:

.

Then , , , .

Then a stays at , , , (say). The unique semicircle, with center at the origin, perpendicular to the one on 1z must have a radius tangent to the radius of the other. The right triangle formed by the abscissa and the perpendicular radii has hypotenuse of length . Since is the radius of the semicircle on 1z, the common perpendicular sought has radius-square

.

The four hyperbolic motions that produced z above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius to yield the unique hyperbolic line perpendicular to both ultraparallels p and q.