[alogo] Envelope of tangent-cuts from a conic to another conic

Here we consider three conics c1, c2, c3 belonging to the same family of conics I(c1, c2), generated by the first two of them. Tangents drawn from points P of conic c1 to c2, c3, cut again c1 respectively at two points A, B. The envelope of line AB is another conic c4, which belongs also to family I(c1,c2).

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

To determine conic c4 apply Desargues involution theorem on line e = AB. By that theorem the pairs of intersection points of e with the members of the family (like pair (A,B)) are in involution, consequently the members of the family I(c1,c2) passing through the fixed points E, G of the involution are tangent there to e. Points E, G being conjugate with respect to any pair (A,B) of intersection points of e with members of I(c1,c2) are related by the quadratic transformation defined by the family I(c1,c2). Consequently all the polars of the members with respect to G pass through E and vice-versa.
The previous remark implies the correctness of the construction of the tangent points E, G from the given data. By moving P on c1, one of the two tangent conics, c4 say, remains constant and e is steadily tangent to this conic.
To prove this one can use the theorem of Hart (see Hart_Lemma.html ). This theorem is usually applied to prove geometrically Poncelet's great theorem. In fact the exercise here is a generalization of that theorem stated for triangles inscribed in c1 (like ABP). If conics c2, c3, c4 coincide, then triangles ABP are inscribed in c1 and circumscribed in c2, and this for all positions of P on the conic c1. Thus, in that case we have an instance of Poncelet's theorem for triangles (the theorem is more general valid for polygons).
Note that the conic c4 tangent to line e for all positions of P on c1 is distinguished from c5 by the fact that lines PE, AD and BC intersect at a point K.

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