[alogo] 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).

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

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.

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