IncircleConjugate

The dual conic of the incircle

This is a particular case from the duality mentioned in TriangleProjectivitiesPlay.html and explained in TriangleConics.html . The conic c under consideration is the dual circumscribed to triangle ABC, when the inscribed conic coincides with the incircle. Several facts are related to this conic (could be called the Soddy conic of the triangle, see [2] below), some of them listed below.

[1] Conic c can be obtained as the image of the circumcircle of the equilateral A₀B₀C₀ through the projectivity H mapping the vertices A₀, B₀, C₀ to A, B, C correspondingly and the center M₀ of the equilateral to the Gergonne point G. These requirements completely determine H, which then maps the incircle of the equilateral to the incircle of ABC. The point named x in the cited reference is here the Gergonne point G and the corresponding trilinear polar is denoted here by L. L coincides with the Gergonne line of the triangle (naming it after Oldknow).
[2] The conic c can be elliptic, parabolic or hyperbolic, depending on the shape of the triangle. Its focal points Z₁, Z₂ are on line (f) passing through G and I. Z₁, Z₂ coincide with the inner/outer Soddy points of the triangle.
[3] Line f is orthogonal to L (called the Soddy line of the triangle). f coincides with the Brocard axis of the contact triangle A_tB_tC_t with vertices at the contact points of incircle with the sides of ABC.
[4] Line L coincides with the Lemoine axis of triangle A_tB_tC_t and carries, among others, the triangle-center X(241), which coincides with X(237) relative to A_tB_tC_t.
[5] Line f carries several triangle centers. For example: the De Longchamps point X(20), X(77), X(170), X(269), X(279), X(347), X(357), X(358), X(390).
[6] Consider another equilateral A₁B₁C₁ resulting by rotating A₀B₀C₀ about its center. Then apply H to get triangle A'B'C'. This triangle shares with ABC several geometric characteristics such as:
[6.1] Same incircle, and conjugate conic, same Gergonne point G, incenter I, lines f and L.
[6.2] Triangle centers of A'B'C' lying on f either remain the same or move on line f. In particular the Soddy points of all triangles A'B'C' are the same.
[6.3] Triangle centers of A'B'C' lying on L either remain fixed or move on line L.
[7] The circumcenters of A'B'C' describe another conic c', not belonging to the family generated by the incircle and its conjugate conic c, sharing though with c one axis.

References

Oldknow, A. The Euler-Gergonne-Soddy triangle of a triangle The Am. Math. Monthly vol. 103(1996) pp. 319-329

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The dual conic of the incircle

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References