[1] For D varying on d, lines B'C' envelope a conic (e), which is tangent to the sides a,b,c of triangle ABC.

[2] Line B'C' and the trilinear polar B*C* of D intersect at a point A* on line BC. Lines C'B* and B'C* intersect also on line BC at a point A

[3] The contact points of (e) with sides b, c are their intersection points with line d.

[4] The contact point of (e) with side a is the intersection point of side a with AS, where S is the tripole of d.

[5] The variable tangents B'C' have tripoles D' lying on the perspectrix of e.

These are generalizations of the analogous properties for hyperbolas studied in HyperbolaPropertyParallels.html . This general setting can be reduced to the aforementioned one by a projective transformation F fixing lines a, b, c and sending line d to infinity. Then the envelope of C'B' is mapped to an hyperbola, hence the stated result. The result on the tangency is related to the fact that B

For some complementary remarks on the various incidence relations of the figure see InscribedConicAsEnvelope.html .

CrossRatioLines.html

HyperbolaPropertyParallels.html

InscribedConicAsEnvelope.html

IncircleTangents.html

Trapezium.html

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