Holditch theorem and Steiner formula for the planar hyperbolic motions (Q5902123)

The authors prove a generalized Holditch theorem in the hyperbolic plane: If a line segment \(XY\) undergoes a closed hyperbolic motion, such that both \(X\) and \(Y\) sweep the same curve \(k\), the area between \(k\) and the trajectory of a point \(Z\) on the segment \(XY\) equals \(\frac{1}{2}\delta ab\) where \(\delta\) is the total rotation angle of the motion and \(a\) and \(b\) are the lengths of the segments \(XZ\) and \(YZ\), respectively. The proof is based on the hyperbolic Steiner area formula for the trajectory of a point. All computations are carried out in the calculus of hyperbolic numbers for the description of planar hyperbolic motions.