Tangent_Cuts_Envelope

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).

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.

Envelope of tangent-cuts from a conic to another conic

See Also