[alogo] 1. The simplest inconic

The simplest inconic is the incircle of an equilateral triangle. In the file IncircleTangents.html it is proved in detail that the tangents to this incircle are the trilinear polars tr(P0) of points P0 lying at the line at infinity L0 (a second proof is given in IsogonalGeneralized.html ).
The line at infinity in turn is the trilinear polar of the center O of the equilateral. Thus, we have a procedure determining the incircle by taking repeatedly trilinear polars of points:
- Start with the system (A0B0C0, O) of an equilateral and its center O.
- Take the trilinear polar L0=tr(O) of O with respect to triangle A0B0C0 which is the line at infinity.
- For every point P0 on L0 take the trilinear polar tr(P0) of P0 with respect to A0B0C0. All these lines envelope the incircle of A0B0C0.

[alogo] 2. The general inconic

If c is an inconic (i.e. inscribed ) of the triangle ABC, then Brianchon's theorem for triangles (see Brianchon3.html ) implies that the lines joining contact points on the sides with opposite vertices meet at a point P. This implies that the inconic (c) can be obtained from the incircle 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 inscribed in the triangle. Since F preserves lines the contact points {A1,B1,C1} of c0 are mapped to the contact points {A',B',C'} of c'. Thus, both conics c and c' pass through these points and have there the same tangents consequently they coincide.
Since projectivities respect cross ratios of points on a line, the line at infinity L0 is mapped by F to the trilinear polar tr(P) and trilinear polars of points P0 on L0 are mapped to trilinear polars of points Q=F(P0) on tr(P). Since all lines tr(P0) envelope c0, the trilinear polars tr(Q) of points Q on tr(P) will also envelope conic c.

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This property of inconics can be thus formulated:
Theorem Every conic inscribed in a triangle defines a point P (called its perspector) which is the intersection point of the lines joining contact points with opposite vertices. The conic is the envelope of trilinear polars tr(Q) of points Q lying on the trilinear polar tr(P) of P (called the perspectrix of the conic).

[alogo] 3. The incircle

The tangents to the incircle of a triangle are the trilinear polars of points on its Gergonne line (tripolar tr(K) of the Gergonne point K).

The proof follows from the very definition of the Gergonne point K as intersection point of the lines joining the contact points with opposite vertices. Thus apply the previous general results to this special case.

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Note that the incircle is the circumcircle of triangle A'B'C' of contact points with the sides of ABC. The Gergonne point with respect to ABC coincides with the symmedian point K with respect to A'B'C'. On the other side the trilinear polar of K with respect to triangles ABC and A'B'C' is the same line (see TrilinearPolar.html ). Since the trilinear polar of the symmedian point with respect to the circumcircle is the Lemoine axis of the triangle, we conclude:

Corollary The tangents to the circumcircle are the trilinear polars of points on the Lemoine axis of the triangle.

See CircumconicsTangents.html for a discussion on an analogous generation of the circumconics of a triangle.

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