Stated as the dual to Maclaurin's conic generation theorem.

If a triangle XYZ varies so that all its vertices {X,Y,Z} are on fixed sides {BC, CA, AB} respectively and two of its sides {XY,XZ} pass through two fixed points {P,Q} respectively, then the third side YZ envelopes a conic.

The proof follows from the corresponding dual in Maclaurin.html (see also Braikenridge.html ).

The conic is seen easily to be tangent to the given lines {AB, AC} and the lines constructible from the data.

{PQ, PB, QC}.

There is a more general form of the theorem referred to as the "Braikenridge theorem" according to which 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 (see Braikenridge.html ).

Braikenridge.htm

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