This is a generalization of the inverse of Maclaurin's theorem (see Maclaurin.html ).

In the figure below it is illustrated in the case of a quadrangle ABCD. The four vertices of the quadrangle move correspondingly on four fixed lines. Also sides {AB, BC, CD} pass correspondingly from four fixed points {P, Q, R}. Then the free side DA envelopes a conic c.

The proof of the theorem is a combination of the Chasles-Steiner generation of conics and the group property of Moebius transformations.

In fact, the correspondence F

b = (k

Analogously are described by corresponding Moebius transformations the maps F

Then points {A,D} of the free line of the quadrangle are related by the composition map.

F: A->D, with F = F

This is again a Moebius transformation and the theorem follows, as claimed, from the Chasles-Steiner argument (see Chasles_Steiner.html ).

Obviously the proof generalizes to arbitrary n-gons.

Chasles_Steiner.html

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