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In this paper we study Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. We show that some collineations of M(A) preserve cross-ratio and thus establish a relation between harmonicity and harmonic position. [ABSTRACT FROM AUTHOR]

In this paper, we study on Moufang–Klingenberg planes $\mathbf{M}(\mathcal{A})$ defined over a local alternative ring $\mathcal{A}$ of dual numbers. We get relations between some special 6-figures and its ratios. Also, we construct a relation between the solvability of the special quadric equation in $\mathcal{A}$ and the existence of the special 6-figure in $\mathbf{M}(\mathcal{A})$. Moreover, we give a construction which provides an algorithm for "squaring" and "cubing" a cross-ratio of points in $\mathbf{M}(\mathcal{A})$. [ABSTRACT FROM AUTHOR]