TrianglesGivenPivotCanonical

Triangle with given pivot

The following construction conforms to the recipe discussed in TrianglesGivenPivot.html , by which,
given a conic c and a point O not lying on c, one constructs triangles ABC such that O is the pivot of the given conic with respect to O.
Here is carried out the construction of a distinguished triangle such that one side of it is parallel to the polar p_O.
1) Construct the diameter OC through O, which is conjugate to the direction of the polar p_O.
2) If this diameter intersects the conic at {C,D} and the polar p_O at F, then E = F(C,D), i.e. the harmonic conjugate of F w.r. to {C,D} is a point on the side of the triangle.
3) Draw parallel from E to p_O intersecting the conic at {A,B} and defining the triangle ABC.

The figure displays also the conic c' (hyperbola) which in the aforementioned reference is used in the construction of the pivoting triangle.

Triangle with given pivot

See Also