Tangent4Lines

All conics tangent to four lines

Problem: Construct all conics tangent to four given lines determined by the sides of a quadrilateral q = A₁A₂A₃A₄ (may be convex or not).

Analysis: A first crucial fact is that the diagonals and the lines joining opposite tangency points X₁X₂ and Y₁Y₂ intersect at the same point C₃. C₁, C₂, C₃ are the intersection points of opposite sides of q. A second crucial fact is that C₃ is the pole of the line C₁C₂ w.r. to every conic inscribed in q. A third crucial fact is that X, Y are harmonic conjugate w.r. to B₁₃, B₂₄. These three facts suffice to complete a recipe for the construction of all conics tangent to the sides of quadrilateral q.
Construction: Start with an arbitrary point X on line C₁C₂. Construct the conjugate harmonic Y of X w.r. to B₁₃, B₂₄. Join X, Y with C₃ and define points X₁, X₂, Y₁, Y₂ through the intersections with the sides of q. Join Y₁ with C₁ and define the intersection point W with line X₁X₂. Define the harmonic conjugate Z of Y₁ w.r. to C₁ and W (on line C₁Y₁). The five points {X₁,X₂,Y₁,Y₂, Z} are on a conic tangent to the sides of q.
Remark Note that the centers O of all these conics lie on the Newton line of the quadrilateral i.e. the line O₁O₂ joining the middles of the diagonals (see Newton.html ). Note also that this line is parallel to the symmetry axis of the unique parabola tangent to the sides of q. In that case O is the point at infinity of line O₁O₂. For an analytic proof of the remark see: Aubert, P & Papelier, G. Exercices de Geometrie Analytique Paris, Librairie Vuibert, 1966, p. 249.

All conics tangent to four lines

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