Braikenridge

Braikenridge theorem

If a polygon with n sides is restricted to have all its vertices on fixed lines and n-1 of its sides pass through corresponding n-1 fixed points then its free side envelopes a conic.
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_P: A->B is described in line coordinates of the corresponding lines through a Moebius transformation.
                                                          b = (k₁a+l₁)/(m₁a+n₁).
Analogously are described by corresponding Moebius transformations the maps F_Q:B->C and  F_R: C->D.
Then points {A,D} of the free line of the quadrangle are related by the composition map.
                                                 F: A->D, with F = F_R*F_Q*F_P  (* denoting composition).
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.

Braikenridge theorem

See Also