CircumconicsTangents

1. The simplest circumconic

The simplest circumconic is the circumcircle c₀ of an equilateral triangle A₀B₀C₀. The trilinear polar of the center O of the equilateral is the line at infinity. The circumcircle c₀ is the incircle of the anticomplementary triangle A₁B₁C₁ of the equilateral. The trilinear polars tr₁(P₀) of points P₀ at infinity with respect to A₁B₁C₁ envelope c₀.
This is discussed in InconicsTangents.html . There is also shown how this generalizes to arbitrary conics inscribed in an arbitrary triangle.

2. The general circumconic

If c is a circumconic (i.e. circumscribed ) of the triangle ABC, then Pascal's theorem for triangles (see PascalOnTriangles.html ) implies that the lines joining the vertices the tangential triangle A''B''C'' of the conic (formed by the tangents of the conic at the vertices of ABC) with opposite vertices of the triangle pass through a point P. This implies that the circumconic (c) can be obtained from the circumcircle c₀ of the equilateral by an appropriate projectivity F.
In fact, define F by the requirements to map the vertices of the equilateral {A₀,B₀,C₀} to {A,B,C} and its center (or centroid) O to point P. Having defined F one can apply it to (c₀) and obtain its image c'=F(c₀) which is a conic circumscribing triangle ABC. Since F preserves lines and contacts and cross ratios the anticomplementary triangle A₁B₁C₁ of ABC maps to triangle A''B''C'' of the harmonic associates of P with respect to ABC. Conic c'=F(c₀) passes through the vertices of ABC and has there the same tangents with c. Hence the two conics coincide.
The line at infinity L₀ is mapped by F to the trilinear polar tr(P) and trilinear polars with respect to A₁B₁Cof points P₀ on L₀ are mapped to trilinear polars with respect to A''B''C'' of points Q=F(P₀) on tr(P). Since all lines tr₁(P₀) envelope c₀ (see IncircleTangents.html ), the trilinear polars tr''(Q) of points Q on tr(P) will also envelope conic c.

Symbols tr, tr₁ and tr'' used above denote the operation of taking the tripolar of a point with respect to triangles ABC and A₁B₁C₁ and A''B''C'' respectively.

3. Generation by tripoles

Circumconic c is also the locus of tripolars tr(L) of lines L through point P. This is noticed in IsogonalGeneralized.html for the case of the equilateral. The general case results by taking the image by F (or using coordinates). The figure indicates the relation between the tripole tr(PQ) of line PQ and the contact point Q' of the tangent tr''(Q). The two points together with P are collinear. See the file TriangleConics.html for an alternative discussion using coordinates.

References

Yiu, P. GeometryNotes020402 (pdf). http://www.math.fau.edu/yiu/GeometryNotes020402.pdf

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1. The simplest circumconic

2. The general circumconic

3. Generation by tripoles

See Also

References