We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.
For an arbitrary triangle ABC and an integer n we define points Dn, En, Fn on the sides BC, CA, AB respectively, in such a manner that |AC|n|AB|n=|CDn||BDn|,|AB|n|BC|n=|AEn||CEn|,|BC|n|AC|n=|BFn||AFn|. $$\matrix{{{{\left| {AC} \right|^n } \over {\left| {AB} \right|^n }} = {{\left| {CD_n } \right|} \over {\left| {BD_n } \right|}},} \hfill & {{{\left| {AB} \right|^n } \over {\left| {BC} \right|^n }} = {{\left| {AE_n } \right|} \over {\left| {CE_n } \right|}},} \hfill & {{{\left| {BC} \right|^n } \over {\left| {AC} \right|^n }} = {{\left| {BF_n } \right|} \over {\left| {AF_n } \right|}}.}} $$ Cevians ADn, BEn, CFn are said to be the Maneeals of order n. In this paper we discuss some properties of the Maneeals and related objects.
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