[alogo] Maclaurin Dual (Braikenridge theorem)

Given are two points {P,Q} and three lines forming the sides of triangle ABC. For each point X on side BC draw {XP,XQ} intersecting respectively sides {AC,AB} at {Y,Z}. Then line YZ envelopes a conic as X varies on BC.
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.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

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 ).

See Also


Return to Gallery

Produced with EucliDraw©